高等数学方法在中学数学中的应用研究
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摘要
随着新课程改革的不断进行,高等数学的知识在高考所占的比重也越来越大,所以,作为高三教师,就必须认真研究新的课程标准、新的考试大纲,认真研究、分析高中数学中的新知识——高等数学的知识方法在中学数学中的应用问题。(2007年广东省将对新课程改革试验进行验收。)
高等数学是在初等数学的基础上发展起来的.与初等数学有着紧密的联系。许多初等数学无法解答的问题高等数学都给出了解答.因此,帮助学生学会用高等数学的思想、方法为工具,从不同的角度去研究初等数学的问题.这些问题可以是与中学教学内容密切相关但又未能完全解决,而应用所学高等数学知识可以解决的理论、方法问题,也可以是初等数学中己经解决,而运用高等数学的知识,从另一更高的角度重新认识初等数学中重要的概念、理论实质及其背景,还可以借助于高等数学的方法来统一处理和解决初等数学中一些或一类问题(尽管这些问题可以用初等的方法来解决)等等。总之应用高等数学的方法使学生对初等数学的本质,以及与高等数学之间的内在联系,有了深刻的认识。
所以本论文选题的基本内容是高等数学方法在中学数学中的应用研究。主要论述的高等数学的方法有微积分方法、极限思想方法、向量方法、逻辑方法、概率统计方法。为了论述这些方法的应用,精选了大量的高考试题。
本论文以大量、翔实的中学数学的范例为依据,尤其是近几年的高考试题,充分说明了高等数学方法在解决中学数学的相关问题上,具有明显的作用。同时,本论文在借鉴前人所撰文章的精神的基础之上,充分借助多年高三教学经验,将高等数学方法在中学数学中的应用写得更加详细、全面。并且尽可能地使用现有中学数学教材讲到的知识、方法。尤其适合与中学数学同行们互相交流,对指导教学,指明方向、深度有重大的参考和借鉴价值。同时,更可以为广大高中学生阅读,以拓广他们的解题思路,提高知识的应用能力。
本论文详细研究了中学数学、高等数学、现代数学的概念、范畴、关系。能使学生对此三个相关联的概念加以区别。
本论文运用高等数学的先进观点居高临下地分析和处理中学数学内容的问题。主要表现为以下三个方面:一是将高等数学的思想和方法渗透到中学数学中去;二是用具体材料来说明高等数学对中学数学的指导意义;`三是指出中学数学某些难以处理的问题的高等数学背景。
Along with new course reform of continuously carry on, Gao Deng mathematics of the knowledge share at Gao Kao3 Suo3 of specific weight also more and more big, so, Be teacher Gao San, have to earnest research new of course standard, new of examination outline, earnest research, analysis the new information within senior high school mathematics know-Gao Deng3 mathematics of the knowledge method is in mathematics in the high school of application problem..(2007 Guangdong province rightness new course reform experiment carry on acceptance.)
Gao Deng's mathematics is at elementary grade mathematics of foundation top the development get up. Have with elementary grade mathematics close of contact. Much the problem Gao Deng's mathematics that elementary grade mathematics can't answer all answer for understanding. Therefore, help student academic association use Gao Deng mathematics of thought, method is tool, from dissimilarity of angle go to research elementary grade mathematics of problem. These problems can be with high school content of course close related but again can not complete solve, but application the theories, method problem which learn Gao Deng's mathematics knowledge and can solve, can be also in elementary grade mathematics F through solve, but usage Gao Deng mathematics of knowledge, from another a higher angle afresh the understanding elementary grade mathematics importance of the concept, theories be substantial and its background, also can ask for help from Gao Deng mathematics of method to unify processing to reach agreement elementary grade mathematics definitely in some or a type of problem(though these problems can solve with the method of elementary grade) etc..In fine application Gao Deng mathematics of the method make student to elementary grade mathematics of essence, and with Gao Deng's mathematics of of inside contact, have deep of understanding.
So this thesis' choosing the basic contents of is Gao Deng mathematics method's application study in mathematics in high school.Main Gao Deng discuss mathematics of the method have a calculus method, extreme limit thought method, vector method, logic method, all rate statistics method.For discussing these methods of application, carefully selected a great deal of of high examination.
This thesis with a great deal of, thoroughly high school mathematics of example is basis, particularly is the last few years of high examination, full elucidation Gao Deng's mathematics method in solving a high school mathematics of related problem have obvious of function.In the meantime, this thesis is draw lessons from people of the past the foundation of spirit which write a chapter on, full ask for help of several years Gao San teaching experience, Gao Deng's mathematics method is in mathematics in the high school of application write Be getting more detailed, overall.And possibly the usage existing high school mathematics teaching material speak of knowledge, method.Particularly in keeping with go together with high school mathematics mutual exchanges, to instruction teaching, indicate a direction, depth to have graveness of reference with draw lessons from value.Can even is large in the meantime Gao high school student reading, with Tuo wide their solution way of thinking, exaltation knowledge of application ability.
This thesis detailed research high school mathematics, Gao Deng's mathematics, now the concept, category of Algebra, relation.The ability make student three to this related allied of the concept take into differentiation.
This thesis usage Gao Deng mathematics of forerunner the standpoint occupy vantage point ground analysis and processing mathematics contents in the high school of problem. Main performance is the following 3:a is Gao Deng mathematics of thought and the method seep through to go in high school mathematics; Two is a tools and implements body the material come to elucidation Gao Deng mathematics to high school mathematics of instruction meaning; Gao Deng3 being the problem which point out the some and hard processing of high school mathematics mathematics background.
高等数学是在初等数学的基础上发展起来的.与初等数学有着紧密的联系。许多初等数学无法解答的问题高等数学都给出了解答.因此,帮助学生学会用高等数学的思想、方法为工具,从不同的角度去研究初等数学的问题.这些问题可以是与中学教学内容密切相关但又未能完全解决,而应用所学高等数学知识可以解决的理论、方法问题,也可以是初等数学中己经解决,而运用高等数学的知识,从另一更高的角度重新认识初等数学中重要的概念、理论实质及其背景,还可以借助于高等数学的方法来统一处理和解决初等数学中一些或一类问题(尽管这些问题可以用初等的方法来解决)等等。总之应用高等数学的方法使学生对初等数学的本质,以及与高等数学之间的内在联系,有了深刻的认识。
所以本论文选题的基本内容是高等数学方法在中学数学中的应用研究。主要论述的高等数学的方法有微积分方法、极限思想方法、向量方法、逻辑方法、概率统计方法。为了论述这些方法的应用,精选了大量的高考试题。
本论文以大量、翔实的中学数学的范例为依据,尤其是近几年的高考试题,充分说明了高等数学方法在解决中学数学的相关问题上,具有明显的作用。同时,本论文在借鉴前人所撰文章的精神的基础之上,充分借助多年高三教学经验,将高等数学方法在中学数学中的应用写得更加详细、全面。并且尽可能地使用现有中学数学教材讲到的知识、方法。尤其适合与中学数学同行们互相交流,对指导教学,指明方向、深度有重大的参考和借鉴价值。同时,更可以为广大高中学生阅读,以拓广他们的解题思路,提高知识的应用能力。
本论文详细研究了中学数学、高等数学、现代数学的概念、范畴、关系。能使学生对此三个相关联的概念加以区别。
本论文运用高等数学的先进观点居高临下地分析和处理中学数学内容的问题。主要表现为以下三个方面:一是将高等数学的思想和方法渗透到中学数学中去;二是用具体材料来说明高等数学对中学数学的指导意义;`三是指出中学数学某些难以处理的问题的高等数学背景。
Along with new course reform of continuously carry on, Gao Deng mathematics of the knowledge share at Gao Kao3 Suo3 of specific weight also more and more big, so, Be teacher Gao San, have to earnest research new of course standard, new of examination outline, earnest research, analysis the new information within senior high school mathematics know-Gao Deng3 mathematics of the knowledge method is in mathematics in the high school of application problem..(2007 Guangdong province rightness new course reform experiment carry on acceptance.)
Gao Deng's mathematics is at elementary grade mathematics of foundation top the development get up. Have with elementary grade mathematics close of contact. Much the problem Gao Deng's mathematics that elementary grade mathematics can't answer all answer for understanding. Therefore, help student academic association use Gao Deng mathematics of thought, method is tool, from dissimilarity of angle go to research elementary grade mathematics of problem. These problems can be with high school content of course close related but again can not complete solve, but application the theories, method problem which learn Gao Deng's mathematics knowledge and can solve, can be also in elementary grade mathematics F through solve, but usage Gao Deng mathematics of knowledge, from another a higher angle afresh the understanding elementary grade mathematics importance of the concept, theories be substantial and its background, also can ask for help from Gao Deng mathematics of method to unify processing to reach agreement elementary grade mathematics definitely in some or a type of problem(though these problems can solve with the method of elementary grade) etc..In fine application Gao Deng mathematics of the method make student to elementary grade mathematics of essence, and with Gao Deng's mathematics of of inside contact, have deep of understanding.
So this thesis' choosing the basic contents of is Gao Deng mathematics method's application study in mathematics in high school.Main Gao Deng discuss mathematics of the method have a calculus method, extreme limit thought method, vector method, logic method, all rate statistics method.For discussing these methods of application, carefully selected a great deal of of high examination.
This thesis with a great deal of, thoroughly high school mathematics of example is basis, particularly is the last few years of high examination, full elucidation Gao Deng's mathematics method in solving a high school mathematics of related problem have obvious of function.In the meantime, this thesis is draw lessons from people of the past the foundation of spirit which write a chapter on, full ask for help of several years Gao San teaching experience, Gao Deng's mathematics method is in mathematics in the high school of application write Be getting more detailed, overall.And possibly the usage existing high school mathematics teaching material speak of knowledge, method.Particularly in keeping with go together with high school mathematics mutual exchanges, to instruction teaching, indicate a direction, depth to have graveness of reference with draw lessons from value.Can even is large in the meantime Gao high school student reading, with Tuo wide their solution way of thinking, exaltation knowledge of application ability.
This thesis detailed research high school mathematics, Gao Deng's mathematics, now the concept, category of Algebra, relation.The ability make student three to this related allied of the concept take into differentiation.
This thesis usage Gao Deng mathematics of forerunner the standpoint occupy vantage point ground analysis and processing mathematics contents in the high school of problem. Main performance is the following 3:a is Gao Deng mathematics of thought and the method seep through to go in high school mathematics; Two is a tools and implements body the material come to elucidation Gao Deng mathematics to high school mathematics of instruction meaning; Gao Deng3 being the problem which point out the some and hard processing of high school mathematics mathematics background.
引文
[1] 徐利治.数学方法论选讲(第三版)[M].武汉:华中理工大学出版社,2000-1:20-31
[2] 蔡上鹤.数学思想和数学方法[J].中学数学,1997-9:1-4
[3] 张奠宙.数学教育经纬[M].南京:江苏教育出版社, 2002-11-3:87-90
[4] 张奠宙.现代数学与中学数学[M].上海:上海教育出版社,1990:
[5] 刘宏武.新课程的教学方法选择[M].北京:中央民族大学出版社,2004.
[6] [荷兰] 弗赖登塔尔著,陈昌平,唐瑞芬编译.作为教育任务的数学,中小学数学教学论著丛书, 1995-3
[7] 沈钢.高观点下的初等数学概念[M].杭州:浙江大学出版社,2001.
[8] 吕世虎,徐兆亮.从高等数学看中学数学[M].北京:科学出版社,1995.
[9] 胡炳生,吴俊,王佩瑾,孙国汉.现代数学观点下的中学数学[M].北京:高等教育出版社,2001
[10] 李文林.数学史概论[M].高等数学教育出版社, 2002-8
[11] 田载今,李海东。关于高中数学教科书中球的表面积公式推导的一点说明[J].中学数学教学参考,2007(05) :11-12
[12] 杨伊生.课堂教学心理学基础[M].呼和浩特:远方出版社,
[13] 冯凤萍. 谈微积分中的数学思想及其教学[J].边疆经济与文化 , 2004(10)
[14] 刘鹏远, 樊春红. 导数应用点滴[J]. 中学生数学 , 2006(05)
[15] 赵素彩. 导数应用剖析两例[J]. 中学理科 , 2006(07)
[16] 焦景会, 黄安成. 导数应用 6 点[J]. 高中数理化 , 2006(06)
[17] 马若飞. 中学要重视“微积分初步”的教学[J]. 课程.教材.教法 ,1987(12)
[18] 王仁发.高观点下的中学数学——代数学 高等教育出版社 2001-7
[19] 沈广艳.高观点下的中学数学——几何学 (网上课件)
[20] 李云杰.“高观点”下的中学数学的实践与认识〔D〕.福建:福建师范大学学报,2005(3)
[21] 罗琳,彭家麒.高观点下的高考试题〔J〕.数学通讯,2003-9
[22] 裘晓岚.关于高观点下初等数学教育的探讨〔J〕.福建师范大学学报,1999,15(3)
[23] 连春兴.高等数学对中学数学教学作用初探〔J〕.北京教育学院学报,2000-3
[24] 周春荔.数学观与方法论〔M〕.北京:首都师范大学出版社,1996,
[2] 蔡上鹤.数学思想和数学方法[J].中学数学,1997-9:1-4
[3] 张奠宙.数学教育经纬[M].南京:江苏教育出版社, 2002-11-3:87-90
[4] 张奠宙.现代数学与中学数学[M].上海:上海教育出版社,1990:
[5] 刘宏武.新课程的教学方法选择[M].北京:中央民族大学出版社,2004.
[6] [荷兰] 弗赖登塔尔著,陈昌平,唐瑞芬编译.作为教育任务的数学,中小学数学教学论著丛书, 1995-3
[7] 沈钢.高观点下的初等数学概念[M].杭州:浙江大学出版社,2001.
[8] 吕世虎,徐兆亮.从高等数学看中学数学[M].北京:科学出版社,1995.
[9] 胡炳生,吴俊,王佩瑾,孙国汉.现代数学观点下的中学数学[M].北京:高等教育出版社,2001
[10] 李文林.数学史概论[M].高等数学教育出版社, 2002-8
[11] 田载今,李海东。关于高中数学教科书中球的表面积公式推导的一点说明[J].中学数学教学参考,2007(05) :11-12
[12] 杨伊生.课堂教学心理学基础[M].呼和浩特:远方出版社,
[13] 冯凤萍. 谈微积分中的数学思想及其教学[J].边疆经济与文化 , 2004(10)
[14] 刘鹏远, 樊春红. 导数应用点滴[J]. 中学生数学 , 2006(05)
[15] 赵素彩. 导数应用剖析两例[J]. 中学理科 , 2006(07)
[16] 焦景会, 黄安成. 导数应用 6 点[J]. 高中数理化 , 2006(06)
[17] 马若飞. 中学要重视“微积分初步”的教学[J]. 课程.教材.教法 ,1987(12)
[18] 王仁发.高观点下的中学数学——代数学 高等教育出版社 2001-7
[19] 沈广艳.高观点下的中学数学——几何学 (网上课件)
[20] 李云杰.“高观点”下的中学数学的实践与认识〔D〕.福建:福建师范大学学报,2005(3)
[21] 罗琳,彭家麒.高观点下的高考试题〔J〕.数学通讯,2003-9
[22] 裘晓岚.关于高观点下初等数学教育的探讨〔J〕.福建师范大学学报,1999,15(3)
[23] 连春兴.高等数学对中学数学教学作用初探〔J〕.北京教育学院学报,2000-3
[24] 周春荔.数学观与方法论〔M〕.北京:首都师范大学出版社,1996,