合情推理在高中数学探究学习中的应用研究
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摘要
近些年来,随着我国数学课程改革向纵深处的发展,探究学习、合情推理等具有开放性和探索性的教育理念日益受到关注。这些新的教学理念体现了数学教学建设性的方向,其首要表征在于数学创新意识的培养。数学的创造过程与其他科学一样,是经过观察、实验、归纳、类比、推广、限定、猜想、联想等一套自然科学中常用方法发现并经过论证而得到的。2003年,我国教育部颁布的《普通高中数学课程标准(实验)》(下称《标准》),已明确应将探究学习作为一种重要的学习方式。“推理与证明”专题亦出现在《标准》中,教材编写中第一次把合情推理引入高中数学课程。现有资料显示,对数学探究学习及合情推理的理论层面的探讨已日趋完善,然而转化为实践层面的具体行为需要一个过程,而且这个过程比想象的要困难得多。
事实上,有研究表明,在实际数学教学过程中,探究学习与合情推理涉及到诸多无法绕过的问题。例如,数学探究学习在实际应用过程中有以下层面的缺失:(1)数学化。数学学习不应停留于实际操作性的活动,只有通过活动的“内化”,我们才可能发展起一定的数学思维。(2)再创造。探究学习是一个不确定性的再创造发现过程,学习知识只是我们成长历程中必经的一个路径,真正的目的在于创造。(3)学会学习。探究学习一方面是让学生懂得知识的动态过程,另一方面是教会学生如何学习。这些蕴涵在探索过程中的精髓思想却被我们所忽视,严重影响了数学探究学习的有效性。合情推理在实际教学中也有被“窄化”的现象,即教师只是就教材知识所及,将其作为概念教学处理,使合情推理甚至只能“养在深闺人未识”,忽视其作为一种数学推理和方法的广泛应用性,也使学生合情推理能力的培养陷入“死胡同”。郑毓信教授谈到,数学课程改革的三个阶段为激情阶段、困惑阶段、务实阶段,现在该是我们带着激情的信念,用务实的态度解决困惑的时候了!
基于以上问题,本文在尊重、吸收已有研究成果的基础上,借助新课程改革的理念,力求有所突破和创新,试图从以下角度作一有益的尝试,即探寻合情推理与数学探究学习整合研究的意义,以研究合情推理在数学探究学习中的应用为内容,将合情推理作为一种促进学生数学探究学习的手段,并以此为载体达到培养学生合情推理能力的目的。旨在使学生合情推理能力的培养落到实处,也为数学探究学习的有效性提升提供方法论的支持。
本研究无论是理论构建还是实践调查,均取得了一定的研究成果。具体体现在以下几个方面:
(1)合情推理在高中数学探究学习中的作用未得到有效体现。
(2)合情推理在高中数学探究学习中应用的理论构建。
(2)合情推理在高中数学探究学习中应用的实践方案。
With China's deep reform of the mathematics education in primary school and middle junior school, such opening and creative educational conventions as research learning and reasonable illation have been attached increasing importance in recent years. Those new educational conventions incarnate the direction of mathematics education with its prime token as the cultivation of mathematics creativity. The process of creating mathematics is like other sciences, which is the result of a set of scientific methods consisting of observation, experiment, induction, analogy, generalization, restriction, supposition and association. In 2003, The Mathematics Curriculum Criteria in General Senior High School issued by China's Department of Education has clearly listed research learning as an important way, and the subject of "Illation and Testify" has also been listed in "Criteria", in which the reasonable illation has been introduced in the compiling of teaching materials for the first time. Statistics shows that the research on the theory of the research learning and reasonable illation of Mathematics has been complete, while the research on the transformation into practice has a long way to go, and this process is much more difficult than expected.
In fact, researches indicate that, in the practical teaching, research learning and reasonable illation involve in many difficult questions. For example, there exist the following defects in the process of practical mathematics teaching: (1) Mathematicization. That is to say, the learning of mathematics should not be inclined to the practical operation, and only with the internalization, the mathematics thinking can be developed. (2) Re-creation. Research learning is an uncertain re-creative process, in which learning knowledge is just an inevitable way in human beings' growing process, and the real aim is to create. (3) Learn how to learn. For one thing, research learning allows students to know the dynamic process of knowledge, and for another thing, it can teach students how to learn. These vital ways of thinking in the researching process, however, have been neglected, which exerts a great influence on the effectiveness of mathematics research learning. The phenomenon of the narrowalization of the reasonable illation in the practical teaching, that is, teachers just teach the students with the knowledge in the textbooks as a kind of concept. On the contrary, reasonable illation has been greatly neglected as a kind of broad application of mathematics illation and method, which leads to the fact that students cannot make full use of the way of reasonable illation in their learning. Professor Zheng Yu-xin classifies the reform of mathematics into three steps: enthusiasm, confusion and putting it into practice. So, it's the time for us to figure the confusion out with enthusiasm as the conviction and putting it into practice as the attitude.
Based on the above research questions, this thesis attempts to make a tentative research on the significance of the combination of reasonable illation and mathematics research learning from the perspective of a new curriculum reform to make a breakthrough and creation on the basis of the previous studies. And it aims to cultivate students' ability of reasonable illation with the study of the application of reasonable illation in mathematics research learning as the content and reasonable illation as a means of promoting students' mathematics research learning. Finally, this thesis provides a methodological support for the improvement of mathematics research learning's effectiveness.
The present study has made some findings both theoretically and practically which include the following aspects:
(1) The ineffective incarnation of the reasonable illation in mathematics research learning in Senior High School.
(2) The theoretical framework of the application of the reasonable illation in mathematics research learning in Senior High School.
(3) The practical plan of the reasonable illation in mathematics research learning in Senior High School.
事实上,有研究表明,在实际数学教学过程中,探究学习与合情推理涉及到诸多无法绕过的问题。例如,数学探究学习在实际应用过程中有以下层面的缺失:(1)数学化。数学学习不应停留于实际操作性的活动,只有通过活动的“内化”,我们才可能发展起一定的数学思维。(2)再创造。探究学习是一个不确定性的再创造发现过程,学习知识只是我们成长历程中必经的一个路径,真正的目的在于创造。(3)学会学习。探究学习一方面是让学生懂得知识的动态过程,另一方面是教会学生如何学习。这些蕴涵在探索过程中的精髓思想却被我们所忽视,严重影响了数学探究学习的有效性。合情推理在实际教学中也有被“窄化”的现象,即教师只是就教材知识所及,将其作为概念教学处理,使合情推理甚至只能“养在深闺人未识”,忽视其作为一种数学推理和方法的广泛应用性,也使学生合情推理能力的培养陷入“死胡同”。郑毓信教授谈到,数学课程改革的三个阶段为激情阶段、困惑阶段、务实阶段,现在该是我们带着激情的信念,用务实的态度解决困惑的时候了!
基于以上问题,本文在尊重、吸收已有研究成果的基础上,借助新课程改革的理念,力求有所突破和创新,试图从以下角度作一有益的尝试,即探寻合情推理与数学探究学习整合研究的意义,以研究合情推理在数学探究学习中的应用为内容,将合情推理作为一种促进学生数学探究学习的手段,并以此为载体达到培养学生合情推理能力的目的。旨在使学生合情推理能力的培养落到实处,也为数学探究学习的有效性提升提供方法论的支持。
本研究无论是理论构建还是实践调查,均取得了一定的研究成果。具体体现在以下几个方面:
(1)合情推理在高中数学探究学习中的作用未得到有效体现。
(2)合情推理在高中数学探究学习中应用的理论构建。
(2)合情推理在高中数学探究学习中应用的实践方案。
With China's deep reform of the mathematics education in primary school and middle junior school, such opening and creative educational conventions as research learning and reasonable illation have been attached increasing importance in recent years. Those new educational conventions incarnate the direction of mathematics education with its prime token as the cultivation of mathematics creativity. The process of creating mathematics is like other sciences, which is the result of a set of scientific methods consisting of observation, experiment, induction, analogy, generalization, restriction, supposition and association. In 2003, The Mathematics Curriculum Criteria in General Senior High School issued by China's Department of Education has clearly listed research learning as an important way, and the subject of "Illation and Testify" has also been listed in "Criteria", in which the reasonable illation has been introduced in the compiling of teaching materials for the first time. Statistics shows that the research on the theory of the research learning and reasonable illation of Mathematics has been complete, while the research on the transformation into practice has a long way to go, and this process is much more difficult than expected.
In fact, researches indicate that, in the practical teaching, research learning and reasonable illation involve in many difficult questions. For example, there exist the following defects in the process of practical mathematics teaching: (1) Mathematicization. That is to say, the learning of mathematics should not be inclined to the practical operation, and only with the internalization, the mathematics thinking can be developed. (2) Re-creation. Research learning is an uncertain re-creative process, in which learning knowledge is just an inevitable way in human beings' growing process, and the real aim is to create. (3) Learn how to learn. For one thing, research learning allows students to know the dynamic process of knowledge, and for another thing, it can teach students how to learn. These vital ways of thinking in the researching process, however, have been neglected, which exerts a great influence on the effectiveness of mathematics research learning. The phenomenon of the narrowalization of the reasonable illation in the practical teaching, that is, teachers just teach the students with the knowledge in the textbooks as a kind of concept. On the contrary, reasonable illation has been greatly neglected as a kind of broad application of mathematics illation and method, which leads to the fact that students cannot make full use of the way of reasonable illation in their learning. Professor Zheng Yu-xin classifies the reform of mathematics into three steps: enthusiasm, confusion and putting it into practice. So, it's the time for us to figure the confusion out with enthusiasm as the conviction and putting it into practice as the attitude.
Based on the above research questions, this thesis attempts to make a tentative research on the significance of the combination of reasonable illation and mathematics research learning from the perspective of a new curriculum reform to make a breakthrough and creation on the basis of the previous studies. And it aims to cultivate students' ability of reasonable illation with the study of the application of reasonable illation in mathematics research learning as the content and reasonable illation as a means of promoting students' mathematics research learning. Finally, this thesis provides a methodological support for the improvement of mathematics research learning's effectiveness.
The present study has made some findings both theoretically and practically which include the following aspects:
(1) The ineffective incarnation of the reasonable illation in mathematics research learning in Senior High School.
(2) The theoretical framework of the application of the reasonable illation in mathematics research learning in Senior High School.
(3) The practical plan of the reasonable illation in mathematics research learning in Senior High School.
引文
[1]国家数学课程标准研制工作组.普通高中数学课程标准(实验)[M].北京:人民教育出版社,2003:39-45.
[2]普通高中课程标准试验教科书(数学)选修1-2[M].北京:人民教育出版社A版,2007:28-42.
[3]于明华.高中数学合情推理课程内容的研究[D].长春:东北师范大学,2007.
[4]宁连华.数学探究学习研究[D].南京:南京大学,2004.
[5]曹才翰,章建跃.数学教育心理学[M].北京:北京师范大学出版社:2002:48.
[6]范文贵.数学探究学习内涵与策略研究[D].上海:华东师范大学,2004.
[7]涂荣豹.数学教学认识论[M].南京:南京师范大学出版社,2003:109.
[8]徐学福.美国“探究教学”研究30年[J].全球教育展望,2001,(8):25.
[9]徐学福,宋乃庆.20世纪探究教学理论的发展及启示[J].西南师范大学学报(人文社会科学版),2001,(7):7-9.
[10]柴西芹.对探究教学的认识与思考[J].课程·教材·教法,2001,(8):21.
[11]钟启泉编译.现代教学论发展[M].北京:北京教育科学出版社,2006:157-160.
[12]盆益萍.研究性学习实验与探索[M].南宁:广西教育出版社,1999:112-116.
[13]韩万富,李善明.论合情推理在中学数学中的地位与功能[J].北京教育学院学报,2000,(2):15.
[14]王晓辉.发展初中生数学合情推理能力的教学实践思考[D].吉林:东北师范大学,2005.
[15]波利亚.数学与猜想(1,2卷)[M]:北京:科学出版社,1984.
[16]任丽洁.数学课程中的合情推理及其教学研究[D].昆明:云南师范大学,2006.
[17]陈拓.中学数学中合情推理教学的研究[D].北京:首都师范大学,2004.
[18]王海坤,葛莉.数学直觉与合情推理对数学教学的意义[J].数学通报,2005,(1):17.
[19]章建跃,朱文芳.中学数学教育心理学[M].北京:北京教育出社,2001:72-80.
[20]波利亚.怎样解题[M].上海:上海科技教育出版社,2002:7.
[21]波利亚.数学的发现(第二卷).刘景麟等译.呼和浩特:内蒙古人民出版社.1981:108.
[22]江丕权.关于教与学[J].中国教育学刊,2003,(10):12.
[23]任樟辉.数学思维论[M].南宁:广西教育出版社,2003:48.
[24]李伟红.高中数学教学中运用合情推理模式的探索[D].济南:山东师范大学,2007:4-5.
[25]郑毓信.数学方法论[M].南宁:广西教育出版社,2003.
[26]杨惠娟.新课标下重析波利亚的合情推理思想[J].数学通报,2006,(2):36.
[27]宁连华.数学探究教学设计的层次及其原则分析[J].中学数学杂志(初中),2007,(6):2-4.
[28]罗增儒.数学解题学引论[M].西安:陕西师范大学出版社,2001:122-123.
[29]陈柏良.寻找适合学生的教学设计[J].中学数学教学参考(高中),2007,(7):13-14.
[30]徐成华.初中数学教育中的合情推理能力培养初探[D].武汉:华中师范大学数学系,2006.
[31]陈琦,刘儒德.当代教育心理学[M].北京:北京师范大学出版社,2004.
[32]方金秋.合情推理与数学学习[J].数学教师,1994,(8):26.
[33]郁建辉.对数学教育中合情推理方法教育认识[J].数学教育学报,1995,(5).
[34]张国银.波利亚的合情推理与学生数学素质培养[J].甘肃教育,2003,(5).
[35]徐斌艳.数学课程与教学论[M].杭州:浙江教育出版社,2003.
[36]Amold Ostebee.Mathematics by Experiment:Plausible Reasoning in the 21st Century[J].The American Mathematical Monthly,Aug/Sep2004.Vol.111,Iss.
[37]Wendy B Sanchez.Helping Students Make Sense of Mathematics[J].The MathematicsTeacher,Dec 2006/Jan 2007.Vol.100,Iss.5.
[38]涂荣豹.试论反思性数学学习[J].数学教育学报,2000,(11).
[39]周春荔.数学创新意识与智力开发[M].北京:首都师范大学出版社,2000.
[40]钱佩玲,邵光华.数学思想方法与中学数学[M].北京:北京师范大学出版社,1999.
[41]徐沥泉.教学·研究·发现——MM方式演绎[M].北京:科学出版社,2003.
[42]章建跃.中学生数学学科自我监控能力[M].上海:华东师范大学出社,2003.
[43]英国国家数学课程标[EB/OL].http://www.shuxue.net/ArticleShow2.asp?AticleID=742.2006-6-22.
[2]普通高中课程标准试验教科书(数学)选修1-2[M].北京:人民教育出版社A版,2007:28-42.
[3]于明华.高中数学合情推理课程内容的研究[D].长春:东北师范大学,2007.
[4]宁连华.数学探究学习研究[D].南京:南京大学,2004.
[5]曹才翰,章建跃.数学教育心理学[M].北京:北京师范大学出版社:2002:48.
[6]范文贵.数学探究学习内涵与策略研究[D].上海:华东师范大学,2004.
[7]涂荣豹.数学教学认识论[M].南京:南京师范大学出版社,2003:109.
[8]徐学福.美国“探究教学”研究30年[J].全球教育展望,2001,(8):25.
[9]徐学福,宋乃庆.20世纪探究教学理论的发展及启示[J].西南师范大学学报(人文社会科学版),2001,(7):7-9.
[10]柴西芹.对探究教学的认识与思考[J].课程·教材·教法,2001,(8):21.
[11]钟启泉编译.现代教学论发展[M].北京:北京教育科学出版社,2006:157-160.
[12]盆益萍.研究性学习实验与探索[M].南宁:广西教育出版社,1999:112-116.
[13]韩万富,李善明.论合情推理在中学数学中的地位与功能[J].北京教育学院学报,2000,(2):15.
[14]王晓辉.发展初中生数学合情推理能力的教学实践思考[D].吉林:东北师范大学,2005.
[15]波利亚.数学与猜想(1,2卷)[M]:北京:科学出版社,1984.
[16]任丽洁.数学课程中的合情推理及其教学研究[D].昆明:云南师范大学,2006.
[17]陈拓.中学数学中合情推理教学的研究[D].北京:首都师范大学,2004.
[18]王海坤,葛莉.数学直觉与合情推理对数学教学的意义[J].数学通报,2005,(1):17.
[19]章建跃,朱文芳.中学数学教育心理学[M].北京:北京教育出社,2001:72-80.
[20]波利亚.怎样解题[M].上海:上海科技教育出版社,2002:7.
[21]波利亚.数学的发现(第二卷).刘景麟等译.呼和浩特:内蒙古人民出版社.1981:108.
[22]江丕权.关于教与学[J].中国教育学刊,2003,(10):12.
[23]任樟辉.数学思维论[M].南宁:广西教育出版社,2003:48.
[24]李伟红.高中数学教学中运用合情推理模式的探索[D].济南:山东师范大学,2007:4-5.
[25]郑毓信.数学方法论[M].南宁:广西教育出版社,2003.
[26]杨惠娟.新课标下重析波利亚的合情推理思想[J].数学通报,2006,(2):36.
[27]宁连华.数学探究教学设计的层次及其原则分析[J].中学数学杂志(初中),2007,(6):2-4.
[28]罗增儒.数学解题学引论[M].西安:陕西师范大学出版社,2001:122-123.
[29]陈柏良.寻找适合学生的教学设计[J].中学数学教学参考(高中),2007,(7):13-14.
[30]徐成华.初中数学教育中的合情推理能力培养初探[D].武汉:华中师范大学数学系,2006.
[31]陈琦,刘儒德.当代教育心理学[M].北京:北京师范大学出版社,2004.
[32]方金秋.合情推理与数学学习[J].数学教师,1994,(8):26.
[33]郁建辉.对数学教育中合情推理方法教育认识[J].数学教育学报,1995,(5).
[34]张国银.波利亚的合情推理与学生数学素质培养[J].甘肃教育,2003,(5).
[35]徐斌艳.数学课程与教学论[M].杭州:浙江教育出版社,2003.
[36]Amold Ostebee.Mathematics by Experiment:Plausible Reasoning in the 21st Century[J].The American Mathematical Monthly,Aug/Sep2004.Vol.111,Iss.
[37]Wendy B Sanchez.Helping Students Make Sense of Mathematics[J].The MathematicsTeacher,Dec 2006/Jan 2007.Vol.100,Iss.5.
[38]涂荣豹.试论反思性数学学习[J].数学教育学报,2000,(11).
[39]周春荔.数学创新意识与智力开发[M].北京:首都师范大学出版社,2000.
[40]钱佩玲,邵光华.数学思想方法与中学数学[M].北京:北京师范大学出版社,1999.
[41]徐沥泉.教学·研究·发现——MM方式演绎[M].北京:科学出版社,2003.
[42]章建跃.中学生数学学科自我监控能力[M].上海:华东师范大学出社,2003.
[43]英国国家数学课程标[EB/OL].http://www.shuxue.net/ArticleShow2.asp?AticleID=742.2006-6-22.