高中数学“算法初步”的教学研究
摘要
随着计算机技术的迅猛发展,算法思想已经成为现代人应具备的一种数学素养.我国新课程标准中不但将算法安排为高中必修内容,还要求把算法思想贯穿到整个高中数学学习的全过程当中.然而,算法教学在实践中还存在各种各样的困难.本文在整理前人研究成果的基础上,对目前算法教学的现状进行了调查和分析,期望能对算法教学带来一些启示
本文首先通过对“算法初步”教学现状的调查,发现在实际教学中存在以下问题：(1)新课程培训不够.在调查中只有少数教师对《普通高中数学课程标准(实验)》(以下简称为《新课标》)中算法内容的教学要求有很清楚的了解,部分教师对算法的一些内容还很生疏.(2)教学难点把握不到位.在调查中,教师认为学生的学习难点主要集中在循环结构及其流程图、循环语句、变量与赋值等内容,而学生认为自己在循环结构及其流程图、实际问题中算法流程图的设计、循环语句等方面存在障碍.(3)教学内容过于单一.学生期待能增加更多的钾力游戏、实际生活问题的算法、计算机语言程序设计和古代数学中的算法问题等.(4)计算机水平有待提高.教师的计算机基础普遍比较薄弱,大部分教师都不能利用计算机实现算法.(5)教材内容有待改善.北师大版教材中部分内容过于零散,概括性与逻辑性不强；算法语句书写不规范,与实际计算机程序还存在很大差距,无法在计算机上实现；部分习题偏难,学生难以理解和掌握.
其次,本文结合程序性知识学习理论、学习策略理论和多元智力理论,分析了《新课标》中对“算法初步”的课程目标、教学内容与要求,并对北师人版教材进行了较为详细的分析和评价,认为该教材的设计编排还存在一些问题.比如,第一节中的算法案例文字叙述过于冗长,不够典型；“选择结构”与“变量与赋值”中例题的设置不够合理,不符合学生的认知规律；算法语句书写不规范等.
再次,本文就算法教学中的一个典型案例进行了深入的分析,探讨了学生在算法学习过程中的思维误区.结果发现学生对于算法流程图的各个框图的理解还存在偏差,对循环结构中循环终止条件的理解还不到位.
最后,本文结合前面的研究和分析,提出了针对算法教学的五点建议.第一,教师应转变观念,注重对学生算法思想的培养.第二,教师要加强对新课程改革相关理论的培训和学习.第三,教师要加强算法相关知识的学习,扩展知识面.不仅掌握有关算法的数学史知识,最好还能掌握一定的计算机程序设计知识.第四,在教学过程中,教师应重视算法案例的选择和使用.第五,鼓励学生交流合作,重视学生能力的培养.
With the rapid development of computer technology, the idea of algorithm has become a mathematical literacy that the modern people should have. The new curriculum standard of China not only arranges the algorithm for the high school mathematics required content, but also requires that the algorithm idea is throughout the whole process of the entire high school mathematics learning. But there are still some difficulties in the practices of algorithm teaching. So, on the basis of sorting the predecessors' achievements, this paper investigate and analyze the status of the algorithm teaching, in order to bring some revelations about algorithm teaching.
First of all, through the investigation of "The preliminary of algorithm" teaching situation, this paper found some problems in the practical teaching of "The preliminary of algorithm" as follows: (1)The new curriculum training is insufficient. Only a minority of teachers have very clear understanding about the new curriculum standers, partial teachers don't understand some content of the algorithm. (2)The grasp of teaching difficulties is inaccurate. In the survey, teachers believe that students' learning difficulties are mainly concentrated in loop structure and its flowcharts, loop statement, variable and assignment, etc. but students think their learning obstacles are loop structure and its flowcharts, algorithm flowcharts of design in the practical problems, looping statements, etc. (3) The teaching content is so onefold, need to be abundant. Students expect to more add intelligence game, algorithm of the real problem, computer language of program design and the algorithm in the ancient mathematical problems, etc. (4) Computer level should be further enhanced. The teacher's basic knowledge of computer are usually quite weak, most of the teachers can't use the computer to operation algorithm. (5) The textbook's contents should be improved. Some content of the textbook(the edition of Beijing Normal University press)are too scattered, generalization and logic is not strong; It's not provided the writing standards of the algorithm statements, some algorithm statements are far from the actual computer program statements, they cannot be operated in the computer; Some exercises are so difficult for students to understand and master.
Secondly, combined with the theory of procedural knowledge learning, learning strategy theory and multiple intelligence theory, this paper analyzes course goal, teaching content and requirement of "algorithm of the preliminary" in the new curriculum standard, and then minutely analyzes and evaluates the textbooks of Beijing Normal University press. It found that there exist some problems. For example, in the first quarter, literal account of the algorithm are too long, not typically enough; The examples in the "selecting structure" and "variable and assignment" are unreasonable, not accord with the students' cognitive law; The written standers of the algorithm statements is nonstandard.
And then, this paper in-depth analyzes a typical case of the algorithm teaching, and discusses the students'thought erroneous zone in the process of learning algorithm. Then, it found that there are some misunderstandings about each block diagram of the algorithm flowchart, and the termination conditions of circular structure.
Finally, based on the previous study and analysis, this paper proposed five suggestions about algorithm teaching. First, teachers should change minds, and pay attention to the development of students' arithmetic idea. Second, teachers should strengthen the training and learning of the new curriculum reform related theory. Third, teachers need to strengthen the algorithm learning and expand their knowledge, not only master relevant knowledge of the mathematics history about algorithms, but also master certain knowledge of computer programming. Fourth, in the teaching process, teachers should attach importance to the choice and apply of algorithm case. Fifth, teacher should encourage students to exchange and cooperation; attach to develop student's ability.
本文首先通过对“算法初步”教学现状的调查,发现在实际教学中存在以下问题：(1)新课程培训不够.在调查中只有少数教师对《普通高中数学课程标准(实验)》(以下简称为《新课标》)中算法内容的教学要求有很清楚的了解,部分教师对算法的一些内容还很生疏.(2)教学难点把握不到位.在调查中,教师认为学生的学习难点主要集中在循环结构及其流程图、循环语句、变量与赋值等内容,而学生认为自己在循环结构及其流程图、实际问题中算法流程图的设计、循环语句等方面存在障碍.(3)教学内容过于单一.学生期待能增加更多的钾力游戏、实际生活问题的算法、计算机语言程序设计和古代数学中的算法问题等.(4)计算机水平有待提高.教师的计算机基础普遍比较薄弱,大部分教师都不能利用计算机实现算法.(5)教材内容有待改善.北师大版教材中部分内容过于零散,概括性与逻辑性不强；算法语句书写不规范,与实际计算机程序还存在很大差距,无法在计算机上实现；部分习题偏难,学生难以理解和掌握.
其次,本文结合程序性知识学习理论、学习策略理论和多元智力理论,分析了《新课标》中对“算法初步”的课程目标、教学内容与要求,并对北师人版教材进行了较为详细的分析和评价,认为该教材的设计编排还存在一些问题.比如,第一节中的算法案例文字叙述过于冗长,不够典型；“选择结构”与“变量与赋值”中例题的设置不够合理,不符合学生的认知规律；算法语句书写不规范等.
再次,本文就算法教学中的一个典型案例进行了深入的分析,探讨了学生在算法学习过程中的思维误区.结果发现学生对于算法流程图的各个框图的理解还存在偏差,对循环结构中循环终止条件的理解还不到位.
最后,本文结合前面的研究和分析,提出了针对算法教学的五点建议.第一,教师应转变观念,注重对学生算法思想的培养.第二,教师要加强对新课程改革相关理论的培训和学习.第三,教师要加强算法相关知识的学习,扩展知识面.不仅掌握有关算法的数学史知识,最好还能掌握一定的计算机程序设计知识.第四,在教学过程中,教师应重视算法案例的选择和使用.第五,鼓励学生交流合作,重视学生能力的培养.
With the rapid development of computer technology, the idea of algorithm has become a mathematical literacy that the modern people should have. The new curriculum standard of China not only arranges the algorithm for the high school mathematics required content, but also requires that the algorithm idea is throughout the whole process of the entire high school mathematics learning. But there are still some difficulties in the practices of algorithm teaching. So, on the basis of sorting the predecessors' achievements, this paper investigate and analyze the status of the algorithm teaching, in order to bring some revelations about algorithm teaching.
First of all, through the investigation of "The preliminary of algorithm" teaching situation, this paper found some problems in the practical teaching of "The preliminary of algorithm" as follows: (1)The new curriculum training is insufficient. Only a minority of teachers have very clear understanding about the new curriculum standers, partial teachers don't understand some content of the algorithm. (2)The grasp of teaching difficulties is inaccurate. In the survey, teachers believe that students' learning difficulties are mainly concentrated in loop structure and its flowcharts, loop statement, variable and assignment, etc. but students think their learning obstacles are loop structure and its flowcharts, algorithm flowcharts of design in the practical problems, looping statements, etc. (3) The teaching content is so onefold, need to be abundant. Students expect to more add intelligence game, algorithm of the real problem, computer language of program design and the algorithm in the ancient mathematical problems, etc. (4) Computer level should be further enhanced. The teacher's basic knowledge of computer are usually quite weak, most of the teachers can't use the computer to operation algorithm. (5) The textbook's contents should be improved. Some content of the textbook(the edition of Beijing Normal University press)are too scattered, generalization and logic is not strong; It's not provided the writing standards of the algorithm statements, some algorithm statements are far from the actual computer program statements, they cannot be operated in the computer; Some exercises are so difficult for students to understand and master.
Secondly, combined with the theory of procedural knowledge learning, learning strategy theory and multiple intelligence theory, this paper analyzes course goal, teaching content and requirement of "algorithm of the preliminary" in the new curriculum standard, and then minutely analyzes and evaluates the textbooks of Beijing Normal University press. It found that there exist some problems. For example, in the first quarter, literal account of the algorithm are too long, not typically enough; The examples in the "selecting structure" and "variable and assignment" are unreasonable, not accord with the students' cognitive law; The written standers of the algorithm statements is nonstandard.
And then, this paper in-depth analyzes a typical case of the algorithm teaching, and discusses the students'thought erroneous zone in the process of learning algorithm. Then, it found that there are some misunderstandings about each block diagram of the algorithm flowchart, and the termination conditions of circular structure.
Finally, based on the previous study and analysis, this paper proposed five suggestions about algorithm teaching. First, teachers should change minds, and pay attention to the development of students' arithmetic idea. Second, teachers should strengthen the training and learning of the new curriculum reform related theory. Third, teachers need to strengthen the algorithm learning and expand their knowledge, not only master relevant knowledge of the mathematics history about algorithms, but also master certain knowledge of computer programming. Fourth, in the teaching process, teachers should attach importance to the choice and apply of algorithm case. Fifth, teacher should encourage students to exchange and cooperation; attach to develop student's ability.
引文
[1]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,2003：24-27,116-120,115.
[2]鲍建生,周超.数学学习的心理基础与过程[M].上海：上海教育出版社,2009：256-257.
[3]陈国芳.高中数学新课程中算法教学现状的调查与分析[D].长春：东北师范大学,2006.
[4]卢盛华.如何避免将数学与算法等同起来——一项关于数学观的访谈研究[J].数学教育学报,2007.16(4)：45-47.
[5]聂力.数学算法研究与教学分析[D].济南：山东师范大学,2004.
[6]程璞.高中数学新课程算法教学的研究[D].南京：南京师范大学,2008.
[7]周丽.基于图形化的编程的高中算法教学研究[D].上海：上海师范大学,2008.
[8]伍春兰,曾跃进.高中数学算法初步教学的若干建议[J].数学通报,2009.48(1)： 22-24.
[9]吴凯彬.突出本质注重结构——对算法教学的若干思考[J].数学通报,2005.44(12)：8-10.
[10]彭翕成.《超级画板》帮你教算法(1)～(4)[J].数学通讯,2007.15：13-15、2007.17：6-8、2001.9：12-14、2007.21：12-13.
[11]费泰生.用BASIC语言实现算法[J].数学通讯,2004.15：5-7、2004.19：4-5、2004.21：4-6、2004.23：8-9
[12]史炳星,王桂霞算.法初步[M].北京：高等教育出版社,2005.
[13]朱文芳.数学教学中提倡“算法多样化”的几点认识[J].数学通报,2003(4)：18-20.
[14]李建华.算法及其教育价值[J].数学教育学报,2004.13(3)：46-47.
[15]费泰生.算法初步(1)——几种常见算法[J].数学通讯,2005.3：5-6.
[16]费泰生.算法初步(2)——几种数学问题的算法[J].数学通讯,2001.7：8-10
[17]廖运章.高中课程标准教科书算法内容设计的比较[J].数学教育学报,2007,16(3)： 83-87.
[18]费泰生.“新教材解读”之(6)——算法初步[J].数学通讯,20091(下半月)：2-5.
[19]朱莹.高中数学算法教学研究[D].长沙：湖南师范大学,2008.
[20]胡学贵.高中“算法初步”不同版本教科书内容及其实施的对比研究——以人教A版、苏教版、北师版为例[D].长春：东北师范大学,2008.
[21]谢丽.高中算法教学的探究[D].上海：华东师范大学,2009.
[22]彭爱辉.高中数学课程中的算法以及教学设计研究[D].贵阳：贵州师范大学,2006.
[23]童玲.高中算法初步的教学研究[D].武汉：华中师范大学,2008.
[24]王惠春.对普通高中学生算大思想的调查与教学研究[D].上海：华东师范大学,2006.
[25]Christopher Clapham.Oxford Concise Dictionary of Mathematics [M] (Second Edition).New York:Oxford University Press,1996:5.
[26]Udi Manber(美).黄林鹏,谢瑾奎,陆首博译.算法引论——一种创造性方法[M].北京：电子工业出版社,2010：1.
[27]沈以淡.简明数学词典[M].北京：北京理工大学出版社,2003：295.
[28]数学课程标准研制组.普通高中数学课程标准(实验)解读[M].南京：江苏教育出版社,2004：16.
[29]Michael T. Goodrich Roberto Tamassia(美).算法分析与设计[M].北京：人民邮电出版社,2006：2.
[30]李文铭.数学史简明教程[M].西安：陕西师范大学出版社,2008：76,79-83.
[31]郭书春,刘钝(校点).算经十书[G].沈阳：辽宁教育出版社,1998.
[32]孙宏安.中国古代数学思想[M].大连：大连理工出版社,2008.
[33]教育部.教育部关于印发《关于开展基础教育新课程试验推广工作的意见》的通知http://www.ncet. gov.cn/zcwj/Show Article.asp?ArticleID=59.
[34]莫雷.教育心理学[M].北京：教育科学出版社,2007：54-55.
[35]陈琦,刘儒德.当代教育心理学[M].北京：北京师范大学出版社,2007.
[36]钟启泉,崔允漷,张华.为了中华民族的复兴为了每位学生的发展——《基础教育课程改革纲要(试行)》解读[M].上海：华东师范大学出版社,2001.
[37]严建生,王尚志.普通高中课程标准实验教科书数学3(必修)[M](第五版)北京：北京师范大学出版社,2008：73-115,168-171.
[38]Knuth D E. Algorithmic thinking and Mathematical Thinking[J]. American Mathematical Monthly,1985(92):170-181.
[39]Sanders, Walter. The Teaching and Learning of Algorithms in School Mathematics [M]. Teaching Children Mathematics,1999(1).
[40]Carroll W M. Algorithms in Everyday Mathematics[J]. Teaching Children Mathematics,1997(7).
[2]鲍建生,周超.数学学习的心理基础与过程[M].上海：上海教育出版社,2009：256-257.
[3]陈国芳.高中数学新课程中算法教学现状的调查与分析[D].长春：东北师范大学,2006.
[4]卢盛华.如何避免将数学与算法等同起来——一项关于数学观的访谈研究[J].数学教育学报,2007.16(4)：45-47.
[5]聂力.数学算法研究与教学分析[D].济南：山东师范大学,2004.
[6]程璞.高中数学新课程算法教学的研究[D].南京：南京师范大学,2008.
[7]周丽.基于图形化的编程的高中算法教学研究[D].上海：上海师范大学,2008.
[8]伍春兰,曾跃进.高中数学算法初步教学的若干建议[J].数学通报,2009.48(1)： 22-24.
[9]吴凯彬.突出本质注重结构——对算法教学的若干思考[J].数学通报,2005.44(12)：8-10.
[10]彭翕成.《超级画板》帮你教算法(1)～(4)[J].数学通讯,2007.15：13-15、2007.17：6-8、2001.9：12-14、2007.21：12-13.
[11]费泰生.用BASIC语言实现算法[J].数学通讯,2004.15：5-7、2004.19：4-5、2004.21：4-6、2004.23：8-9
[12]史炳星,王桂霞算.法初步[M].北京：高等教育出版社,2005.
[13]朱文芳.数学教学中提倡“算法多样化”的几点认识[J].数学通报,2003(4)：18-20.
[14]李建华.算法及其教育价值[J].数学教育学报,2004.13(3)：46-47.
[15]费泰生.算法初步(1)——几种常见算法[J].数学通讯,2005.3：5-6.
[16]费泰生.算法初步(2)——几种数学问题的算法[J].数学通讯,2001.7：8-10
[17]廖运章.高中课程标准教科书算法内容设计的比较[J].数学教育学报,2007,16(3)： 83-87.
[18]费泰生.“新教材解读”之(6)——算法初步[J].数学通讯,20091(下半月)：2-5.
[19]朱莹.高中数学算法教学研究[D].长沙：湖南师范大学,2008.
[20]胡学贵.高中“算法初步”不同版本教科书内容及其实施的对比研究——以人教A版、苏教版、北师版为例[D].长春：东北师范大学,2008.
[21]谢丽.高中算法教学的探究[D].上海：华东师范大学,2009.
[22]彭爱辉.高中数学课程中的算法以及教学设计研究[D].贵阳：贵州师范大学,2006.
[23]童玲.高中算法初步的教学研究[D].武汉：华中师范大学,2008.
[24]王惠春.对普通高中学生算大思想的调查与教学研究[D].上海：华东师范大学,2006.
[25]Christopher Clapham.Oxford Concise Dictionary of Mathematics [M] (Second Edition).New York:Oxford University Press,1996:5.
[26]Udi Manber(美).黄林鹏,谢瑾奎,陆首博译.算法引论——一种创造性方法[M].北京：电子工业出版社,2010：1.
[27]沈以淡.简明数学词典[M].北京：北京理工大学出版社,2003：295.
[28]数学课程标准研制组.普通高中数学课程标准(实验)解读[M].南京：江苏教育出版社,2004：16.
[29]Michael T. Goodrich Roberto Tamassia(美).算法分析与设计[M].北京：人民邮电出版社,2006：2.
[30]李文铭.数学史简明教程[M].西安：陕西师范大学出版社,2008：76,79-83.
[31]郭书春,刘钝(校点).算经十书[G].沈阳：辽宁教育出版社,1998.
[32]孙宏安.中国古代数学思想[M].大连：大连理工出版社,2008.
[33]教育部.教育部关于印发《关于开展基础教育新课程试验推广工作的意见》的通知http://www.ncet. gov.cn/zcwj/Show Article.asp?ArticleID=59.
[34]莫雷.教育心理学[M].北京：教育科学出版社,2007：54-55.
[35]陈琦,刘儒德.当代教育心理学[M].北京：北京师范大学出版社,2007.
[36]钟启泉,崔允漷,张华.为了中华民族的复兴为了每位学生的发展——《基础教育课程改革纲要(试行)》解读[M].上海：华东师范大学出版社,2001.
[37]严建生,王尚志.普通高中课程标准实验教科书数学3(必修)[M](第五版)北京：北京师范大学出版社,2008：73-115,168-171.
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