一元微积分概念教学的设计研究
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摘要
大众化背景下,大学生入学时的能力普遍降低,学生层次越来越不均衡,这已经成为世界高等教育面临的一个主要问题。另一方面,基础教育课程改革的推进使得中学的课程设置发生了巨大的变化,这种变化也对大学的课程设置提出了新的要求。大众化教育以及高中课改的背景使得大学微积分教学中的问题日益突出,很多大学生会进行求导、积分运算,但是对概念中蕴含的思想并不理解,对概念间的关系认识模糊。所以,发现学生在微积分概念上的认知困难并进行有针对性的教学设计是微积分教学改革的关键。
本论文以一元微积分作为载体,选取极限、导数、微分、中值定理、定积分等内容作为研究的切入点,研究了2个问题：(1)大学生对微积分中的基本概念具有什么样的概念意象,存在哪些概念误解?(2)如何设计微积分的概念教学,以加深学生对概念的理解,提高其运用基本概念的能力?
本研究构建了微积分概念教学原则,并对一所理工院校大一上学期三个教学班的微积分课程进行了教学设计与教学实验,主要采用了设计研究、问卷调查、访谈、课堂观察、准实验对照等研究方法,有3位教师以及255位学生参加了概念教学班的教学实践。研究包括3个阶段：(1)准备和设计：根据现有文献及教学经验总结出学生所遇到的常见错误与问题以及每个案例教学设计的要点(设计原型),设计出概念的前/后测试卷,对测试时间、教学时间作出安排。(2)教学实践：针对前测中发现的问题,对原有的教学设计(设计原型)进行修正,并实施概念教学。(3)回顾分析：任课教师撰写教学反思,并对概念教学设计原则进行修正;依据修正后的原则,开始下一轮的教学设计。在研究的最后,我们进行了教学设计的效果检验,主要通过三条路径：(1)以具体案例的前后测对比,进行教学班纵向的比较；(2)以学校统一安排的期中期末考试进行横向的比较；(3)在学期末,对学生进行调查,了解学生对概念教学的认可情况。
通过研究得到以下结论：
其一,大学生对微积分基本概念的概念意向是片面的,甚至有些是错误的。(1)在学习极限的定义前,大学生不会用严格的语言来界定极限,有一些同学用静态的观点来看待极限,认为极限就是“n趋于无穷大(x趋于x0)时,数列(函数)等于a”。(2)大多数学生在看到导数时首先想到的是函数曲线在某点切线的斜率；学生主要从斜率的角度来理解导数,而非从变化率的角度来理解。(3)学生对通过导数来求微分这种“操作性的知识”认识深刻,但是对微分的几何意义和线性近似的思想认识存在混乱。(4)部分学生知道定积分是面积,但是不清楚究竟是哪个区域的面积;知道定积分概念中的分割与近似代替的过程,但是部分学生不清楚对哪个量进行分割：一些学生单纯地认为dx是积分号的一部分,而忽略了其“微分”的实际意义。
其二,我们构建了微积分概念教学原则,并进行了相应的教学设计与教学实验。微积分概念教学原则如下：(1)通过本原性(历史上的,本质的)问题引入数学概念,借助历史发展阐述数学概念；(2)借助几何直观或生活中的直观例子帮助同学理解概念；(3)注重概念间关系的阐述。针对前测中的问题,每个案例的设计重点如下：极限的教学设计重在通过直观的方式帮助同学熟悉、理解并会运用形式化的语言;导数的教学设计重在阐明概念所蕴含的“变化率”思想；微分的设计重点在于突出概念间的联系,帮助学生在头脑中形成概念图；中值定理的设计重点在于通过历史上的定理形式来让学生体会到概念的严格化过程：定积分是过程性概念的典型代表,其设计要点在于在教学中帮助学生将定积分的概念解压缩,从而将定积分概念迁移到未知情境中。
研究的创新之处在于：在国内首先比较系统地研究了学生对一元微积分基本概念的理解,并剖析了学生的概念意象；针对这些概念意象与学生的概念误解进行了教学设计与为期一个学期的教学实践。研究呈现了微积分概念教学的原始设计、对学生概念意象及概念误解的调查、教学设计的修正、教学设计的实施、教学效果反馈的全过程,其理论意义在于为微积分教学研究提供实证性的依据,为后续研究的开展做一些基础性的工作。实践价值在于可帮助大学教师了解学生的概念理解情况,为教师提供具体的教学策略和教学设计参考,也可为大学的教材编写者提供素材。
In the context of mass higher education, the ability of college freshmen are generally in lower level than before, and increasingly uneven. It has become a major problem facing the world of higher education. On the other hand, reform on basic education has made tremendous change in the secondary school curriculum. This change also put forward new requirements for the university setting. Mass education and the high school curriculum cause problems increasingly on Calculus teaching in university. Many college students can do simple works on derivative and calculus, but can not understand the idea behind the concept, and as a result, usually have fuzzy understanding of the relationship between concepts. Therefore, how to find the cognitive difficulties of the students on the concepts of calculus and to make the targeted instructional design is the key to the reform of the teaching on Calculus.
This research, selecting cases on limit, derivative, differential, Mean Value Theorem, definite integral, focuses on the two questions:(1) What concept images and misunderstandings do college students have on the basic concepts in calculus?(2) How to design the teaching of calculus concepts can help students understand the concepts, and improve their ability to apply basic concepts?
This research, based on the construction of principles on concept instruction in calculus, presents a case study on calculus course in three classes of a college of science and engineering freshmen by carrying out the teaching design and teaching experiment. Research methods such as design research, questionnaires, interview, classroom observation, control of quasi-experimental are adopted. There are3teachers and255students participated in the practice. The study consists of three phases.1) Preparation and design:based on literature review and teaching experience we sum up the common errors and problems encountered by students, and the main points in concept instruction. Then we design test volumes for before/after the teaching, and make arrangements of test time, teaching time.(2) Teaching practice: according to the problems found in the pre-test, we correct the original instructional design (prototype) and implement the concept instruction.(3) Review:teachers write teaching reflection on the concept instruction design principles and make second correction; based on the revised principles, the next round of instructional design is carried on. At the end of the study, we conduct a test of the effect of instructional design, primarily through three paths:firstly, making longitudinal comparison among the testing classes according the pre-test and post-test comparison of the specific case; secondly, using midterm and final exams in school uniform arrangements for horizontal comparison; thirdly, at the end of the semester, making the student survey to know students'recognition of the concept instruction.
Based on the findings of this study, the following conclusions can be drawn:
Firstly, college students'concept image of the fundamental concepts of calculus is one-sided, and some even wrong.(1) Before learning, students can not define the limit by correct words. Some students even take a static point of view, as limit is "when n tends to infinity (tend), the number of series (function) is equal to a."(2) Most of the students usually think of the function curve slope of the tangent at a point when seeing derivative; students mainly consider the angle of the slope as derivative, rather than from the point of view of the rate of change in understanding the definition.(3) The "operational knowledge" makes deeply impression to students. There is confusion in the profound understanding of the significance of differential geometry and linear approximation.(4) some students know the definite integral is the area, but it is not clear exactly which area of the region; Some know the definite integral concept segmentation and approximation substitute of the process, but do not know which amount is divided; some students simply think dx as part of the integral symbol, while ignoring the practical significance of the "differential".
Secondly, we constructed principles on concept instruction in calculus and the corresponding instructional design and teaching experiments. The principles are as follows:(1) through the primitive (history, essentially) the introduction of mathematical concepts, with the historical development of elaborate mathematical concepts;(2) by means of geometric or intuitive life examples to help students understand the concepts;(3) paying attention to the elaboration of the concept of the relations between them. Based on the problems in the pre-test, the design focus of each case are as follows:Limit focuses on intuitive way to help students learn, understand and use formal language; Derivative on clarifying the thought of "the rate of change"; Differential on the links between prominent concept to help students in the minds of a concept map; Mean Value Theorem on the history of the theorem in the form to help students understand the concept of strict process; Given integral as a typical representative of the process concept, the design point is to help students in teaching the concept of definite integral decompression, which will be fixed integral concept to migrate to an unknown situation.
The research, first in China, systematically studied students'understanding of the fundamental concepts of calculus, and explores students'concept image through a semester teaching practice. The implications of this study for teaching and learning the concept of calculus as well as on research in mathematics education are also discussed. The practical value is that it can help university teachers to know students' conceptual understanding of the situation, to provide teachers with specific teaching strategies and design reference, and also as material for a university textbook writing.
本论文以一元微积分作为载体,选取极限、导数、微分、中值定理、定积分等内容作为研究的切入点,研究了2个问题：(1)大学生对微积分中的基本概念具有什么样的概念意象,存在哪些概念误解?(2)如何设计微积分的概念教学,以加深学生对概念的理解,提高其运用基本概念的能力?
本研究构建了微积分概念教学原则,并对一所理工院校大一上学期三个教学班的微积分课程进行了教学设计与教学实验,主要采用了设计研究、问卷调查、访谈、课堂观察、准实验对照等研究方法,有3位教师以及255位学生参加了概念教学班的教学实践。研究包括3个阶段：(1)准备和设计：根据现有文献及教学经验总结出学生所遇到的常见错误与问题以及每个案例教学设计的要点(设计原型),设计出概念的前/后测试卷,对测试时间、教学时间作出安排。(2)教学实践：针对前测中发现的问题,对原有的教学设计(设计原型)进行修正,并实施概念教学。(3)回顾分析：任课教师撰写教学反思,并对概念教学设计原则进行修正;依据修正后的原则,开始下一轮的教学设计。在研究的最后,我们进行了教学设计的效果检验,主要通过三条路径：(1)以具体案例的前后测对比,进行教学班纵向的比较；(2)以学校统一安排的期中期末考试进行横向的比较；(3)在学期末,对学生进行调查,了解学生对概念教学的认可情况。
通过研究得到以下结论：
其一,大学生对微积分基本概念的概念意向是片面的,甚至有些是错误的。(1)在学习极限的定义前,大学生不会用严格的语言来界定极限,有一些同学用静态的观点来看待极限,认为极限就是“n趋于无穷大(x趋于x0)时,数列(函数)等于a”。(2)大多数学生在看到导数时首先想到的是函数曲线在某点切线的斜率；学生主要从斜率的角度来理解导数,而非从变化率的角度来理解。(3)学生对通过导数来求微分这种“操作性的知识”认识深刻,但是对微分的几何意义和线性近似的思想认识存在混乱。(4)部分学生知道定积分是面积,但是不清楚究竟是哪个区域的面积;知道定积分概念中的分割与近似代替的过程,但是部分学生不清楚对哪个量进行分割：一些学生单纯地认为dx是积分号的一部分,而忽略了其“微分”的实际意义。
其二,我们构建了微积分概念教学原则,并进行了相应的教学设计与教学实验。微积分概念教学原则如下：(1)通过本原性(历史上的,本质的)问题引入数学概念,借助历史发展阐述数学概念；(2)借助几何直观或生活中的直观例子帮助同学理解概念；(3)注重概念间关系的阐述。针对前测中的问题,每个案例的设计重点如下：极限的教学设计重在通过直观的方式帮助同学熟悉、理解并会运用形式化的语言;导数的教学设计重在阐明概念所蕴含的“变化率”思想；微分的设计重点在于突出概念间的联系,帮助学生在头脑中形成概念图；中值定理的设计重点在于通过历史上的定理形式来让学生体会到概念的严格化过程：定积分是过程性概念的典型代表,其设计要点在于在教学中帮助学生将定积分的概念解压缩,从而将定积分概念迁移到未知情境中。
研究的创新之处在于：在国内首先比较系统地研究了学生对一元微积分基本概念的理解,并剖析了学生的概念意象；针对这些概念意象与学生的概念误解进行了教学设计与为期一个学期的教学实践。研究呈现了微积分概念教学的原始设计、对学生概念意象及概念误解的调查、教学设计的修正、教学设计的实施、教学效果反馈的全过程,其理论意义在于为微积分教学研究提供实证性的依据,为后续研究的开展做一些基础性的工作。实践价值在于可帮助大学教师了解学生的概念理解情况,为教师提供具体的教学策略和教学设计参考,也可为大学的教材编写者提供素材。
In the context of mass higher education, the ability of college freshmen are generally in lower level than before, and increasingly uneven. It has become a major problem facing the world of higher education. On the other hand, reform on basic education has made tremendous change in the secondary school curriculum. This change also put forward new requirements for the university setting. Mass education and the high school curriculum cause problems increasingly on Calculus teaching in university. Many college students can do simple works on derivative and calculus, but can not understand the idea behind the concept, and as a result, usually have fuzzy understanding of the relationship between concepts. Therefore, how to find the cognitive difficulties of the students on the concepts of calculus and to make the targeted instructional design is the key to the reform of the teaching on Calculus.
This research, selecting cases on limit, derivative, differential, Mean Value Theorem, definite integral, focuses on the two questions:(1) What concept images and misunderstandings do college students have on the basic concepts in calculus?(2) How to design the teaching of calculus concepts can help students understand the concepts, and improve their ability to apply basic concepts?
This research, based on the construction of principles on concept instruction in calculus, presents a case study on calculus course in three classes of a college of science and engineering freshmen by carrying out the teaching design and teaching experiment. Research methods such as design research, questionnaires, interview, classroom observation, control of quasi-experimental are adopted. There are3teachers and255students participated in the practice. The study consists of three phases.1) Preparation and design:based on literature review and teaching experience we sum up the common errors and problems encountered by students, and the main points in concept instruction. Then we design test volumes for before/after the teaching, and make arrangements of test time, teaching time.(2) Teaching practice: according to the problems found in the pre-test, we correct the original instructional design (prototype) and implement the concept instruction.(3) Review:teachers write teaching reflection on the concept instruction design principles and make second correction; based on the revised principles, the next round of instructional design is carried on. At the end of the study, we conduct a test of the effect of instructional design, primarily through three paths:firstly, making longitudinal comparison among the testing classes according the pre-test and post-test comparison of the specific case; secondly, using midterm and final exams in school uniform arrangements for horizontal comparison; thirdly, at the end of the semester, making the student survey to know students'recognition of the concept instruction.
Based on the findings of this study, the following conclusions can be drawn:
Firstly, college students'concept image of the fundamental concepts of calculus is one-sided, and some even wrong.(1) Before learning, students can not define the limit by correct words. Some students even take a static point of view, as limit is "when n tends to infinity (tend), the number of series (function) is equal to a."(2) Most of the students usually think of the function curve slope of the tangent at a point when seeing derivative; students mainly consider the angle of the slope as derivative, rather than from the point of view of the rate of change in understanding the definition.(3) The "operational knowledge" makes deeply impression to students. There is confusion in the profound understanding of the significance of differential geometry and linear approximation.(4) some students know the definite integral is the area, but it is not clear exactly which area of the region; Some know the definite integral concept segmentation and approximation substitute of the process, but do not know which amount is divided; some students simply think dx as part of the integral symbol, while ignoring the practical significance of the "differential".
Secondly, we constructed principles on concept instruction in calculus and the corresponding instructional design and teaching experiments. The principles are as follows:(1) through the primitive (history, essentially) the introduction of mathematical concepts, with the historical development of elaborate mathematical concepts;(2) by means of geometric or intuitive life examples to help students understand the concepts;(3) paying attention to the elaboration of the concept of the relations between them. Based on the problems in the pre-test, the design focus of each case are as follows:Limit focuses on intuitive way to help students learn, understand and use formal language; Derivative on clarifying the thought of "the rate of change"; Differential on the links between prominent concept to help students in the minds of a concept map; Mean Value Theorem on the history of the theorem in the form to help students understand the concept of strict process; Given integral as a typical representative of the process concept, the design point is to help students in teaching the concept of definite integral decompression, which will be fixed integral concept to migrate to an unknown situation.
The research, first in China, systematically studied students'understanding of the fundamental concepts of calculus, and explores students'concept image through a semester teaching practice. The implications of this study for teaching and learning the concept of calculus as well as on research in mathematics education are also discussed. The practical value is that it can help university teachers to know students' conceptual understanding of the situation, to provide teachers with specific teaching strategies and design reference, and also as material for a university textbook writing.
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