中学物理课程中数学知识的支持性研究
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摘要
在数学与物理学的整个发展历程中,两者总是紧密交织、相互促进的。然而,由于现代课程的分科体系,导致物理与数学课程及其教学互相分离、彼此孤立。虽然很早就有人注意到因分科而造成学科间知识、技能和能力的割裂,并提出了进行数学课程与科学课程整合以协调两者教学的实践与研究,但直到目前仍缺乏系统研究。教学实践表明,中学物理课程学习中存在数学基础不足和数学运用困难问题。然而,在应试教育观念的影响下,国内有关中学物理课程中数学运用问题的研究主要侧重于运用数学知识解决物理问题的技巧和方式、方法,鲜有从协调数理关系的角度进行的研究。
本研究以基础教育课程改革所倡导的“改变学科本位”、“强调学科渗透”思想为指导,通过对物理课程中数学运用情况的宏观调查和数学运用困难原因的微观剖析,层层深入地研究了数学知识对物理课程的支持问题。研究内容为:首先,通过对中学物理课程标准和教材进行的文本调查和内容分析,从客观上获得物理课程中的数学运用情况,然后通过对初、高中物理教师进行广泛而深入的问卷调查,获得了物理课程所需数学知识、物理课程学习存在的数学知识基础不足与滞后、数学知识的运用困难等情况。在此基础上,就问题的解决进行了两项探索性研究,其一,为获得数学知识滞后对物理学习影响的证据,进行了补充有关三角知识的教学实验,以此探讨如何解决数学知识滞后问题;其二,为研究数学运用困难的具体表现及其原因,以函数图象表征物体运动问题作为测试内容,对20名被试进行了口语报告的分析研究,在此基础上尝试探索减少数学运用困难的途径。本研究的基本结论是:
(1)初中物理课程标准要求达到理解以上的物理概念共有8个、物理规律有3个,而高中物理课程标准的要求分别为25个和39个,并且高中物理课程中的有关内容与初中相比,无论在广度、深度还是综合程度上都有很大提高。高中物理教材中定量概念、规律、以及相应习题的数量,分别是初中教材的2倍、4倍、4倍,从这些内容的教学设计所反映数学能力需求来看,高中物理对学生数学建模能力、空间想象能力、数学推理与分析能力、以及数学运算技能的要求较之初中大大提高了。
(2)初中物理课程学习所需数学知识主要涉及数、比例及代数式的运算知识、基本几何知识等12项,需要程度均为“一般”以下。高中物理课程学习所需数学知识涉及数与式、方程、不等式等的运算知识,函数及其图象、函数求极值知识、平面几何、三角函数、解三角形知识等,共46项,大部分知识的需要程度都在“一般”以上,表明物理课程学习所需数学知识的数量、范围、及需要程度随着阶段上升而不断提高。
(3)高中物理学习所需三角学的知识普遍存在不足或滞后,且对物理学习的影响较大。高中数学中的弧度制、用导数求函数极值、排列与组合、数列及其求和,以及初中几何体的三视图、圆的弧-弦-圆心角的关系等数学知识也存在不足或滞后,但对物理学习的影响一般以下。
(4)高中物理学习中存在运用困难的数学知识主要有函数及其图像、解三角形、锐角三角函数、方程、不等式、求函数极值、几何体的三视图等,然而,平面几何、立体几何知识、数与式的运算、三角函数公式、排列组合、数列求和、概率等知识基本没有运用困难。调查显示,所有“运用困难”问题对物理学习的影响程度总体上高于数学基础不足所带来的影响。分析表明,存在数学运用困难的主要原因在于物理情景与数学情景的差异,数学知识在物理情景中的运用还需要经历新的知识建构过程。
(5)教学实验研究发现,对物理学习影响较大的滞后数学知识予以补充性教学,能够显著提高学生的物理成绩,而对物理学习影响一股的滞后数学知识予以补充对提高学生的物理成绩并不显著。研究还发现,补充数学知识对改善女生以及中等程度学生的物理成绩更为有效。
(6)采用口语报告分析方法研究学生解释运动图象问题时发现,学生能够较好地理解运动图象的特征信息与物理概念之间的联系,而在将图象语言与实际运动情景进行联系的任务中表现欠佳。另外,学生倾向于对不同曲线的点与点进行比较以获得个别信息,较少将图象语言转化为数学符号语言,从而不能从整体上把握图象表征的实际意义。进一步分析发现,学生图象应用困难的原因一方面与数学函数教学本身所具有的抽象性有关,另一方面,函数图象的实际运用能力尚需在物理学习中得到进一步发展。
In the developing history of mathematics and physics, the two disciplines have connected and been co-dependent. They, however, were separated and independent owing to the classified modern course. Although some scholars have already noticed the effects of classified course and proposed to integrate mathematics and the science course so as to harmonize the teaching of the two courses. But the system study is lacking. In educational practice, there are some problems during high school physics learning, such as lacking of mathematics basis and the difficulty of using mathematics. Affected by the examination oriented education, the majority of the academic papers in china explore the mathematical implication skills in solving physical problems. A few research the mathematical implication in teaching physics from the perspective of mathematics and physics coordination.
This research conforms to the trends of "not solely working within the discipline" and "emphasis on discipline permeating" which the fundamental education reform promotes. This dissertation, from a macro review of mathematics application in physics course to a micro analysis of the problem of mathematics application, provides a deep research on supporting state of mathematics to physics course. Firstly, we respectively do a textual and contextual analysis on high school Curricular criteria and course books in order to investigate mathematics application in physics course. Then, we do a wide questionnaire on junior and senior high school physics teachers in order to know the actual situation of it and the problems. On the basis of investigation, this dissertation further explores the problems: one, to do a experimental teaching research of adding trigonometry knowledge to see whether lacking of mathematics knowledge needs improvement; two, to do a test on the question of function graph symbolizing object movement and do an analysis on 20 subjects' verbal protocol in order to discovering the way of eliminate the problem of mathematics implication. This research finally obtains the following findings:
1. In junior high school physics course criteria, there are 8 physical concepts and 3 physical rules which are required to understand and learn more of them. While in senior high school physics course criteria, there are respectively 25 and 39. What's more, the complexity of senior physics course improves in a large scale. The quantitative definition, rule and the relative exercises in the senior high school physics course books are respectively 2 times, 4 times, 4 times as much as that in junior high school course books. The teaching designs also reflect that the requirement of mathematics model ability, space imaginative ability, mathematics reasoning and analysis ability, and mathematical operation ability improves a lot compared with that of junior high school physics course book.
2. The needful degree is below "ordinary" for the mathematics knowledge points which are 12 items including number, proportion, formula and geometry in junior high school physics course book; while in senior high school physics course book, there are 46 items which require a degree beyond "ordinary", which shows the mathematics knowledge during physics learning extremely expands.
3. Trigonometry knowledge is universally wanted in senior high school physics learning, which greatly effects physics learning. While some mathematics knowledge in senior high school learning is also lacking, such as circular measure, permutation, combination, ordered series of numbers and some of them in junior high school learning. But these lacks do not badly affect physics learning.
4. The application problems mainly focus on how to solve triangle, acute triangle function, equation, inequation, function and its image, the three-dimensional image of a solid; while such knowledge as, plane geometry, solid geometry, operation of number and equation, trigonometric function equation, permutation and combination, ordered series of numbers sum, probability has no application problems. All these "application problems" affect physics learning in a higher degree than that of mathematical base weakness. According to our analysis, the main reasons of the application problem lie on the difference between physics situation and mathematics situation. The application of mathematical knowledge in physics situation demands a new process of knowledge building.
5. The teaching experiment research finds out that supplementary mathematics knowledge which highly affects physics learning can prominently improve students' physics grades; while supplementary mathematics knowledge which does not affects physics learning so prominently can hardly improve students' physics grades. Supplementary mathematics knowledge can improve female students' and the high students' physics grades more effectively.
6. A research, employing verbal protocol to explore students' ability of interpreting moving graph, finds out that students can easily understand the relationship between the characteristic information of moving graph and its physics concept; while they are not capable of associating graph language with real moving situation. In addition, students tend to do point to point from two graphs comparison and then to gain individual information rather than to convert graph language into mathematical symbol language. They fail to understand the actual significance of a graph in the whole. A further research discovers that this application problem relates to the innate abstractness of teaching mathematics function.
本研究以基础教育课程改革所倡导的“改变学科本位”、“强调学科渗透”思想为指导,通过对物理课程中数学运用情况的宏观调查和数学运用困难原因的微观剖析,层层深入地研究了数学知识对物理课程的支持问题。研究内容为:首先,通过对中学物理课程标准和教材进行的文本调查和内容分析,从客观上获得物理课程中的数学运用情况,然后通过对初、高中物理教师进行广泛而深入的问卷调查,获得了物理课程所需数学知识、物理课程学习存在的数学知识基础不足与滞后、数学知识的运用困难等情况。在此基础上,就问题的解决进行了两项探索性研究,其一,为获得数学知识滞后对物理学习影响的证据,进行了补充有关三角知识的教学实验,以此探讨如何解决数学知识滞后问题;其二,为研究数学运用困难的具体表现及其原因,以函数图象表征物体运动问题作为测试内容,对20名被试进行了口语报告的分析研究,在此基础上尝试探索减少数学运用困难的途径。本研究的基本结论是:
(1)初中物理课程标准要求达到理解以上的物理概念共有8个、物理规律有3个,而高中物理课程标准的要求分别为25个和39个,并且高中物理课程中的有关内容与初中相比,无论在广度、深度还是综合程度上都有很大提高。高中物理教材中定量概念、规律、以及相应习题的数量,分别是初中教材的2倍、4倍、4倍,从这些内容的教学设计所反映数学能力需求来看,高中物理对学生数学建模能力、空间想象能力、数学推理与分析能力、以及数学运算技能的要求较之初中大大提高了。
(2)初中物理课程学习所需数学知识主要涉及数、比例及代数式的运算知识、基本几何知识等12项,需要程度均为“一般”以下。高中物理课程学习所需数学知识涉及数与式、方程、不等式等的运算知识,函数及其图象、函数求极值知识、平面几何、三角函数、解三角形知识等,共46项,大部分知识的需要程度都在“一般”以上,表明物理课程学习所需数学知识的数量、范围、及需要程度随着阶段上升而不断提高。
(3)高中物理学习所需三角学的知识普遍存在不足或滞后,且对物理学习的影响较大。高中数学中的弧度制、用导数求函数极值、排列与组合、数列及其求和,以及初中几何体的三视图、圆的弧-弦-圆心角的关系等数学知识也存在不足或滞后,但对物理学习的影响一般以下。
(4)高中物理学习中存在运用困难的数学知识主要有函数及其图像、解三角形、锐角三角函数、方程、不等式、求函数极值、几何体的三视图等,然而,平面几何、立体几何知识、数与式的运算、三角函数公式、排列组合、数列求和、概率等知识基本没有运用困难。调查显示,所有“运用困难”问题对物理学习的影响程度总体上高于数学基础不足所带来的影响。分析表明,存在数学运用困难的主要原因在于物理情景与数学情景的差异,数学知识在物理情景中的运用还需要经历新的知识建构过程。
(5)教学实验研究发现,对物理学习影响较大的滞后数学知识予以补充性教学,能够显著提高学生的物理成绩,而对物理学习影响一股的滞后数学知识予以补充对提高学生的物理成绩并不显著。研究还发现,补充数学知识对改善女生以及中等程度学生的物理成绩更为有效。
(6)采用口语报告分析方法研究学生解释运动图象问题时发现,学生能够较好地理解运动图象的特征信息与物理概念之间的联系,而在将图象语言与实际运动情景进行联系的任务中表现欠佳。另外,学生倾向于对不同曲线的点与点进行比较以获得个别信息,较少将图象语言转化为数学符号语言,从而不能从整体上把握图象表征的实际意义。进一步分析发现,学生图象应用困难的原因一方面与数学函数教学本身所具有的抽象性有关,另一方面,函数图象的实际运用能力尚需在物理学习中得到进一步发展。
In the developing history of mathematics and physics, the two disciplines have connected and been co-dependent. They, however, were separated and independent owing to the classified modern course. Although some scholars have already noticed the effects of classified course and proposed to integrate mathematics and the science course so as to harmonize the teaching of the two courses. But the system study is lacking. In educational practice, there are some problems during high school physics learning, such as lacking of mathematics basis and the difficulty of using mathematics. Affected by the examination oriented education, the majority of the academic papers in china explore the mathematical implication skills in solving physical problems. A few research the mathematical implication in teaching physics from the perspective of mathematics and physics coordination.
This research conforms to the trends of "not solely working within the discipline" and "emphasis on discipline permeating" which the fundamental education reform promotes. This dissertation, from a macro review of mathematics application in physics course to a micro analysis of the problem of mathematics application, provides a deep research on supporting state of mathematics to physics course. Firstly, we respectively do a textual and contextual analysis on high school Curricular criteria and course books in order to investigate mathematics application in physics course. Then, we do a wide questionnaire on junior and senior high school physics teachers in order to know the actual situation of it and the problems. On the basis of investigation, this dissertation further explores the problems: one, to do a experimental teaching research of adding trigonometry knowledge to see whether lacking of mathematics knowledge needs improvement; two, to do a test on the question of function graph symbolizing object movement and do an analysis on 20 subjects' verbal protocol in order to discovering the way of eliminate the problem of mathematics implication. This research finally obtains the following findings:
1. In junior high school physics course criteria, there are 8 physical concepts and 3 physical rules which are required to understand and learn more of them. While in senior high school physics course criteria, there are respectively 25 and 39. What's more, the complexity of senior physics course improves in a large scale. The quantitative definition, rule and the relative exercises in the senior high school physics course books are respectively 2 times, 4 times, 4 times as much as that in junior high school course books. The teaching designs also reflect that the requirement of mathematics model ability, space imaginative ability, mathematics reasoning and analysis ability, and mathematical operation ability improves a lot compared with that of junior high school physics course book.
2. The needful degree is below "ordinary" for the mathematics knowledge points which are 12 items including number, proportion, formula and geometry in junior high school physics course book; while in senior high school physics course book, there are 46 items which require a degree beyond "ordinary", which shows the mathematics knowledge during physics learning extremely expands.
3. Trigonometry knowledge is universally wanted in senior high school physics learning, which greatly effects physics learning. While some mathematics knowledge in senior high school learning is also lacking, such as circular measure, permutation, combination, ordered series of numbers and some of them in junior high school learning. But these lacks do not badly affect physics learning.
4. The application problems mainly focus on how to solve triangle, acute triangle function, equation, inequation, function and its image, the three-dimensional image of a solid; while such knowledge as, plane geometry, solid geometry, operation of number and equation, trigonometric function equation, permutation and combination, ordered series of numbers sum, probability has no application problems. All these "application problems" affect physics learning in a higher degree than that of mathematical base weakness. According to our analysis, the main reasons of the application problem lie on the difference between physics situation and mathematics situation. The application of mathematical knowledge in physics situation demands a new process of knowledge building.
5. The teaching experiment research finds out that supplementary mathematics knowledge which highly affects physics learning can prominently improve students' physics grades; while supplementary mathematics knowledge which does not affects physics learning so prominently can hardly improve students' physics grades. Supplementary mathematics knowledge can improve female students' and the high students' physics grades more effectively.
6. A research, employing verbal protocol to explore students' ability of interpreting moving graph, finds out that students can easily understand the relationship between the characteristic information of moving graph and its physics concept; while they are not capable of associating graph language with real moving situation. In addition, students tend to do point to point from two graphs comparison and then to gain individual information rather than to convert graph language into mathematical symbol language. They fail to understand the actual significance of a graph in the whole. A further research discovers that this application problem relates to the innate abstractness of teaching mathematics function.
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