对高三学生无限、极限理解的调查研究
摘要
“无限”是人们日常语言中一个常用的词汇，同时“无限”也是数学中一个重要的概念，但“无限”在日常语言中的含义与数学中的含义却是不一致的，这就可能导致学生在学习涉及“无限”的一些数学概念时产生一些错误的理解。本文就是在这个背景下进行的一项调查研究，选取的数学概念是极限概念，调查主要分为以下两部分内容：
一是通过对学生的书面回答和访谈获得的信息来分析学生对“∞”的理解，主要分两方面来进行：一方面是调查学生赋予“∞”的含义；另一方面是调查学生解决?这一类极限的方法，并考察这两者之间的联系。
二是通过学生对函数极限概念与对和的极限的理解来分析他们对动态极限概念的理解模式。通过学生的书面回答，发现学生理解函数极限概念的模式是：极限接近但不可能到达、极限是可到达的、极限是近似值和极限是精确值：通过访谈，发现学生理解和的极限的模式是：极限是非常接近、极限是近似值、极限是精确值和极限小于精确值。其中极限是非常接近但不可能达到是学生理解动态极限概念的普遍模式。
接下来本文分析了影响学生理解动态极限概念的两个因素。一是语言，即动态极限概念中出现的“无限接近”；二是无限，即学生难以区别数学中无限大和有限之间的含义、只接受潜无限而否认真无限。
在论文的最后部分，向教师提出了在极限教学时的一些建议，并指出了论文的不足之处和进一步研究的课题。
The meaning of infinity as a common vocabulary in our everyday life and an important notion in mathematical fields are quite different, which suggest it's difficult for students to understand some mathematical concepts involving infinity. This study focuses on informal notion of limit and infinity hold by 401 senior school students:
1. By investigation and interview, it was found that students understand mainly "" in the lim f(x) as a large number and potential infinity, and the role those understandings
play in their solutions of limit problems s as lim f(x).
2. By investigation and interview students' informal models that invoked in giving meaning to the function limit and sum limit, it was found that the former informal models involves limit an unreachable-. limit as reachable limit as precise value and limit as approximation ,and the latter involves limit as closeness limit as precise value limit as approximation and limit precise value, of these models, limit as closeness and unreachable is most common misunderstanding . These informal models arose and developed depend on informal meanings of technical language and conceptual difficulty of the notion of infinity.
On the basis of findings mentioned above, some advice are given on limit teaching, it is advisable for teachers to pay attention to appropriate expression of infinity. Finally the limitations of this paper and further study about this topic are pointed out.
一是通过对学生的书面回答和访谈获得的信息来分析学生对“∞”的理解，主要分两方面来进行：一方面是调查学生赋予“∞”的含义；另一方面是调查学生解决?这一类极限的方法，并考察这两者之间的联系。
二是通过学生对函数极限概念与对和的极限的理解来分析他们对动态极限概念的理解模式。通过学生的书面回答，发现学生理解函数极限概念的模式是：极限接近但不可能到达、极限是可到达的、极限是近似值和极限是精确值：通过访谈，发现学生理解和的极限的模式是：极限是非常接近、极限是近似值、极限是精确值和极限小于精确值。其中极限是非常接近但不可能达到是学生理解动态极限概念的普遍模式。
接下来本文分析了影响学生理解动态极限概念的两个因素。一是语言，即动态极限概念中出现的“无限接近”；二是无限，即学生难以区别数学中无限大和有限之间的含义、只接受潜无限而否认真无限。
在论文的最后部分，向教师提出了在极限教学时的一些建议，并指出了论文的不足之处和进一步研究的课题。
The meaning of infinity as a common vocabulary in our everyday life and an important notion in mathematical fields are quite different, which suggest it's difficult for students to understand some mathematical concepts involving infinity. This study focuses on informal notion of limit and infinity hold by 401 senior school students:
1. By investigation and interview, it was found that students understand mainly "" in the lim f(x) as a large number and potential infinity, and the role those understandings
play in their solutions of limit problems s as lim f(x).
2. By investigation and interview students' informal models that invoked in giving meaning to the function limit and sum limit, it was found that the former informal models involves limit an unreachable-. limit as reachable limit as precise value and limit as approximation ,and the latter involves limit as closeness limit as precise value limit as approximation and limit precise value, of these models, limit as closeness and unreachable is most common misunderstanding . These informal models arose and developed depend on informal meanings of technical language and conceptual difficulty of the notion of infinity.
On the basis of findings mentioned above, some advice are given on limit teaching, it is advisable for teachers to pay attention to appropriate expression of infinity. Finally the limitations of this paper and further study about this topic are pointed out.
引文
[1] Davis, R. & Vinner, S. (1986). The Notion of Limit: Some Seemingly Unavoidable Misconception Stages. Journal of Mathematical Behavior, 5, 281-303.
[2] Fischbein, E, et al.(1979) The intuition of infinity, Educational Studies in Mathematics 10, 3-40.
[3] Monaghan, J.(1991).Problems with the language of limits. For the learning of Mathmatics, 11(3), 20-24.
[4] Tall, D.O..&Vinner, S.(1981). Concept image and concept definition in mathmatics, with particular reference to limits and continuity. Educational Studiesin mathmatics. 12, 151-159.
[5] Sierpinska, A.(1987) Humanities students and epistemological obstacles related to limits.Educational Studies in Mathmatics. 18, 371-397.
[6] Szydlik, J.E(2000).Mathmatics beliefs and conceptual understanding of the limit of a fucation .Journal for Research in Mathmatics Education.31 (3), 258-276.
[7] Williams, S.R(2001).Predicatoins of the limit concept:an application of repertory grids. Journal for Research in Mathmatics Education. 32(4), 341-367.
[8] Lisa Denise Murphy.students'conceptions of limit http://www.mste.uiuc.edu/Murphy/papers/limitconceptpaper.html
[9] D.A．格劳斯(陈昌平等译)．数学教与学研究手册[M]．上海：上海教育出版社．1999．664-665
[10] [美]D.休斯.哈雷和A.M.克莱逊(胡乃等译)．微积分[M]．北京：高等教育出版社1997
[11] [德]Rolf Biehler(唐瑞芬等译)．数学教学理论是一门科学[M]．上海：上海教育出版社．1998．
[12] 弗赖登塔尔(陈昌平等译)．作为教育任务的数学[M]．上海：上海教育出版社．1995．
[13] [以色列]伊莱．马奥尔(王前等译)无穷之旅[M]．上海：上海教育出版社．2000．
[14] [日]仲田纪夫(丁树深译)无穷的奥秘及其演变[M]．北京：科学出版．2001
[15] [美]G.伽莫夫.(暴永宁译)从一到无穷大-科学中的事实和臆测[M]．上海：科学出版社
[16] 卢介景．无穷统帅-康托尔[M]．山东：山东教育出版社．2001
[17] 郜穗萍．科学处理教材深刻理解数列极限概念．数学通报[J]．2002(1)，24-28．
[18] 人民教育出版社中学数学室编著．全日制普通高级中学教科书(试验本)第三册[M]北京：人民教育出版社．2001．75．
[19] 孙广润．非标准分析概论[M]．北京：北京科学出版社．1995．
[20] 唐瑞芬．李士绮．国际教育展望：数学教育研究评价[M]．上海：上海教育出版社．1996．225-239
[21] 徐利治．数学方法论选讲[M]．武昌：华中工学院出版社．1983．97-117．
[22] 新英汉词典编写组编．新英汉词典[M]．上海：上海译文出版社．1985．648，739，131，393
[23] 魏琴．杨玉东．理解学生学习极限概念的困难[J]中学教研(数学)2003，9
[24] 张奠宙．数学教育研究导引[M]．江苏：江苏教育出版社．1994
[25] 张奠宙．数学的明天[M]．广西：广西教育出版社．130-131
[26] 李士绮．PME：数学教育心理[M]．上海：华东师范大学出版社．2001．65
[27] 姜涛．关于极限概念的ε-语言[J]．数学教育学报．1999．8(3)．99-101．
[28] 常瑞玲．用数学方法论指导极限概念教学的尝试．数学教育学报[J]2000．9(1)．78-80．
[29] 萧治经．熊萍．关于极限概念的非ε语言．数学教育学报[J]．1998．7(2)．86-88．
[30] 钱林．姚林．经济专业高等数学课程改革初探．数学教育学报[J]．2000．9(1)．64-67．
[31] 郭文秀．极限思想的建立．郑州铁路职业技术学院学报[J]．2001．13(2)．52-54．
[32] 谢文暖．素质教育与极限教学．内蒙古教育学院教育学报．[J]1998．4．81-83．
[33] 晨光．小学数学中的无限．数学家的思维方式与小学数学教育之十六
[34] 郇中丹．对师范大学本科数学专业《数学分析》课程改革的几点意见．数学教育学报[J]．2000．9(2)．17-19
[35] 张定强．剖析高等数学结构 提高学生素质．数学教育学报[J]．1996．5(1)．91-95
[36] 王高峡．唐瑞芬．再谈美国的微积分教学改革．数学教育学报[J]．2000．9(4)．69-72。
[37] 洪方权．“数学分析”的难点教学．数学教育学报[J]．1997．6(4)．72-73．
[38] 钱明忠．论高师数学分析课程内容的选择原则．数学教育学报[J]．1999．8(4)．95-98．
[2] Fischbein, E, et al.(1979) The intuition of infinity, Educational Studies in Mathematics 10, 3-40.
[3] Monaghan, J.(1991).Problems with the language of limits. For the learning of Mathmatics, 11(3), 20-24.
[4] Tall, D.O..&Vinner, S.(1981). Concept image and concept definition in mathmatics, with particular reference to limits and continuity. Educational Studiesin mathmatics. 12, 151-159.
[5] Sierpinska, A.(1987) Humanities students and epistemological obstacles related to limits.Educational Studies in Mathmatics. 18, 371-397.
[6] Szydlik, J.E(2000).Mathmatics beliefs and conceptual understanding of the limit of a fucation .Journal for Research in Mathmatics Education.31 (3), 258-276.
[7] Williams, S.R(2001).Predicatoins of the limit concept:an application of repertory grids. Journal for Research in Mathmatics Education. 32(4), 341-367.
[8] Lisa Denise Murphy.students'conceptions of limit http://www.mste.uiuc.edu/Murphy/papers/limitconceptpaper.html
[9] D.A．格劳斯(陈昌平等译)．数学教与学研究手册[M]．上海：上海教育出版社．1999．664-665
[10] [美]D.休斯.哈雷和A.M.克莱逊(胡乃等译)．微积分[M]．北京：高等教育出版社1997
[11] [德]Rolf Biehler(唐瑞芬等译)．数学教学理论是一门科学[M]．上海：上海教育出版社．1998．
[12] 弗赖登塔尔(陈昌平等译)．作为教育任务的数学[M]．上海：上海教育出版社．1995．
[13] [以色列]伊莱．马奥尔(王前等译)无穷之旅[M]．上海：上海教育出版社．2000．
[14] [日]仲田纪夫(丁树深译)无穷的奥秘及其演变[M]．北京：科学出版．2001
[15] [美]G.伽莫夫.(暴永宁译)从一到无穷大-科学中的事实和臆测[M]．上海：科学出版社
[16] 卢介景．无穷统帅-康托尔[M]．山东：山东教育出版社．2001
[17] 郜穗萍．科学处理教材深刻理解数列极限概念．数学通报[J]．2002(1)，24-28．
[18] 人民教育出版社中学数学室编著．全日制普通高级中学教科书(试验本)第三册[M]北京：人民教育出版社．2001．75．
[19] 孙广润．非标准分析概论[M]．北京：北京科学出版社．1995．
[20] 唐瑞芬．李士绮．国际教育展望：数学教育研究评价[M]．上海：上海教育出版社．1996．225-239
[21] 徐利治．数学方法论选讲[M]．武昌：华中工学院出版社．1983．97-117．
[22] 新英汉词典编写组编．新英汉词典[M]．上海：上海译文出版社．1985．648，739，131，393
[23] 魏琴．杨玉东．理解学生学习极限概念的困难[J]中学教研(数学)2003，9
[24] 张奠宙．数学教育研究导引[M]．江苏：江苏教育出版社．1994
[25] 张奠宙．数学的明天[M]．广西：广西教育出版社．130-131
[26] 李士绮．PME：数学教育心理[M]．上海：华东师范大学出版社．2001．65
[27] 姜涛．关于极限概念的ε-语言[J]．数学教育学报．1999．8(3)．99-101．
[28] 常瑞玲．用数学方法论指导极限概念教学的尝试．数学教育学报[J]2000．9(1)．78-80．
[29] 萧治经．熊萍．关于极限概念的非ε语言．数学教育学报[J]．1998．7(2)．86-88．
[30] 钱林．姚林．经济专业高等数学课程改革初探．数学教育学报[J]．2000．9(1)．64-67．
[31] 郭文秀．极限思想的建立．郑州铁路职业技术学院学报[J]．2001．13(2)．52-54．
[32] 谢文暖．素质教育与极限教学．内蒙古教育学院教育学报．[J]1998．4．81-83．
[33] 晨光．小学数学中的无限．数学家的思维方式与小学数学教育之十六
[34] 郇中丹．对师范大学本科数学专业《数学分析》课程改革的几点意见．数学教育学报[J]．2000．9(2)．17-19
[35] 张定强．剖析高等数学结构 提高学生素质．数学教育学报[J]．1996．5(1)．91-95
[36] 王高峡．唐瑞芬．再谈美国的微积分教学改革．数学教育学报[J]．2000．9(4)．69-72。
[37] 洪方权．“数学分析”的难点教学．数学教育学报[J]．1997．6(4)．72-73．
[38] 钱明忠．论高师数学分析课程内容的选择原则．数学教育学报[J]．1999．8(4)．95-98．
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |