基于APOS理论的圆锥曲线概念教学实证研究
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摘要
数学概念是数学的逻辑起点,是学生认知的基础,在数学学习与教学中具有重要地位。长期以来,数学概念教学通常采用概念形成和概念同化的方式进行,而大多数教师习惯使用概念同化方式的教学,这种教学方式过程比较简明,能使学生比较直接的学习概念,因此被称为“学生获得数学概念的最基本的方式”。然而概念同化教学方式偏重于概念的逻辑结构教学,一个相当普遍的倾向是将知识发生过程的教学一带而过,接着很快便转入教师对各种题型的反复介绍和学生对大量题目的模仿和练习,学生通过课堂教学获得的概念却无法加以运用,从而造成很大一部分中等生及学困生由于对概念理解不深而导致解题时错误屡屡发生。传统的数学概念教学方式导致了一系列的消极后果,与当前我国基础教育倡导的“以学生发展为本”的基本理念是背道而驰的。新课标提出:“对于数学概念教学必须反璞归真,揭示数学概念的形成过程,让学生从概念的现实原形、概念的抽象过程、数学思想的指导作用、形式表述和符号化的运用等多方面理解一个数学概念,使之符合学生主动建构的教育原理。”因此,为了促进学生的发展,我们必须从传统的数学概念教与学模式中走出来。
美国教育学家杜宾斯基(Ed Dubinsky)等人在数学教育研究实践中提出的APOS理论,对数学概念教学很有启示。该理论集中于对特定学习内容——数学概念学习过程的研究,提出学生学习数学概念要经过“活动”、“过程”、“对象”、“图式”四个阶段。APOS理论强调把概念寓于现实社会背景中,学生通过活动亲身经历、体验与现实的联系,从中经历完整的学习过程,用数学方法组织和建立概念。这样建立起来的概念具有丰富的内涵,其中包含着概念的现实原型、概念的抽象过程、思想方法和概念的形式化等,而达到建构数学概念的目的,这与新课标所提倡的学生主动建构是相一致的,顺应了当前数学教育的改革。
本研究在探讨和分析APOS理论的基础上,把APOS理论应用于圆锥曲线概念教学,进一步检验APOS理论概念教学模式应用于数学概念教学的实施效果。研究分为理论研究与教学实证研究两部分。理论研究为教学实证研究奠定了基础,教学实证研究也使得理论研究成果得到更好的诠释,两者相互促进,是我们研究的基本思路。在对国内外的相关文献进行综述的基础上,基于建构主义理论和图式理论的研究,我们对APOS理论的四个阶段进行剖析,探讨APOS理论概念教学设计和实施过程,为开展数学概念教学研究和实践研究提供理论支撑和教学策略;实证研究的对象为柳州市一中高二年级的两个平行班学生共131人,在将近一个学期的调查研究中,从实验的准备阶段到实验的全过程,通过问卷调查、访谈、课堂观察、测试分析等方式,运用统计分析的工具对相关数据进行分析,并结合课堂教学的实际,分析在日常教学中实施APOS理论概念教学的可行性、教学实施的困难和效果。重点是对实验班和对照班学生的数学成绩、数学兴趣、数学探究意识进行比较研究,以及对同一教学内容采取APOS理论概念教学模式和传统概念教学模式的教学过程的比较分析。
教学实证研究的结果较好的切合了理论研究,实证研究的结果表明:运用APOS理论概念教学模式进行数学概念教学激发了学生参与教学的心理机制,激发了学习的热情,提高学生学习数学的兴趣,加深他们对圆锥曲线概念的理解,数学成绩有所提高;APOS理论概念教学模式给学生提供了自主探究、发现问题和解决问题的空间,有助于学生探究意识的加强,初步验证了APOS理论概念教学模式实践的可行性和实施的有效性。
根据在实际教学中实施APOS理论概念教学模式的分析研究,本文最后提出在实施APOS理论概念教学过程中需注意的几点建议:APOS理论的四个阶段是一个相对连续的过程,概念在大脑中建立期间任一阶段都是不可缺少的;创设情境并不是概念教学的最终目的,概念教学不能只停留在活动(操作)层面,而对其他阶段草草收场;数学概念由“过程”到“对象”的建立需经多次反复,循序渐进;对象、图式阶段是数学概念在头脑中建立的长远之计,二者可循环上升;APOS理论的四个阶段并非一定体现在一堂数学课当中,也不是每一课都必须遍历四个阶段,它适用于数学概念在学生头脑中建立的一段时期,并不局限于某一堂课。
Mathematical concept is the logical beginning of maths, the basis for students’cognition. It has an important position in mathematics study and the teaching. Since long ago, mathematics concepts teaching have usually been suggested in the way of Concept Form and Concept Assimilation. Most teachers adopt Concept Assimilation. This kind of teaching process is relatively simple and enable students to learn concepts more directly, and it is called " themost essential way for students to obtain mathematical concepts ". However, this teaching method emphases on teaching the logical structure of concepts. A quite universal tendency is not to take the process of teaching and knowledge seriously.Then various types of exercises are introduced by teachers and students have to do a large number of imitation and practice. Students are unable to utilize the concepts which they obtain through the classroom instruction actually isunable to perform to utilize, thus causing many average students and students who have difficulty in learning to make mistakes when solving problems because of misunderstanding the concepts. The traditional teaching method of the mathematical concept has led to a series of negative consequences. It runs counter to " to the development of students" , the basic idea of elementary education put forward by our country . It’s advocated in the current curriculum standard that mathematical concepts teaching must be reverted to simplicity, reveals the formation of mathematical concepts, thus enabling students to multidimensionally understand the concepts from the prototype to abstract process, the guiding role of mathematical thinking, the use of symbols and forms of expression, which, conforms to the education principle of constructing the construction on their own initiative. Therefore, in order to promote the development of students, we must shift the concept teaching from the traditional model.
American education experts Grosso (Ed Dubinsky) and so on, proposed APOS Theory in mathematics education research practice, which is very enlightening to concept teaching. The theory focuses on the specific contents, that is the learning process of the mathematical concept and proposes four stages to learn mathematics concepts, that is“activity", "process", "object", and "scheme". After the students learn mathematical concepts to "activities" and "process" and "targets", "plans" four stages. APOS theory stressed that the concept resides in real social backgrounds. Students experience its links with reality through activities, experience the integrity of the study process, and use methodology to establish the concept. Concepts based on this method contains a rich connotation, which contains the concept’s prototype, abstract process, thinking method and formalization and so on. The goal of constrcting the mathematical concepts is achieved. It is consistent with“students taking the initiative”advocated in the New Curriculum Standard complies with the current mathematics education reform.
Based on discussing and analyzing the APOS theory, the research applies the APOS theory to the conic curve concept teaching and further examines the implementation effects of the APOS theory concept educational mode being applied to tomathematics concept teaching. The research consists of the fundamental research and teaching empirical study. The fundamental research lays the foundation for teaching empirical study, and the latter also makes a better interpretation for the former. They support each other, which is our basic idea. On the basis of the related literatures, and based on the schema theory of constructivism theory, we analyse the four-phases of APOS theory, and explore the APOS theory concept teaching design and the implementation process, so as to provide theoretical support and teaching strategies for the development of the mathematics concept teaching research and the practical research. Empirical research covers 131 Senior 2 students in two classes in Liuzhou No 1 Middle School. In six months of research, from the preparation to the entire process, through questionnaire surveys, interviews, classroom observation, test analysis, analyze the data by using statistical analysis as a tool, combine with the actual classroom teaching, analyse feasibility and effectiveness or difficulty in implementing APOS theoretical concepts in the regular teaching. The key point is to compare students’mathematics results, mathematics interest, mathematical awareness of exploration between key class and the tested classes, as well as compare and analyse the teaching processes on the same teaching content between the two teaching modes, the APOS theory concepteducational mode and the traditional concept educational mode.
Empirical research findings better fits the theoretical study. The results show that math concept teaching based on inspires students to participate in teaching and learning, stimulate students’interest in mathematics, enhance their understanding of the concept of conic curve, and do better in math. The APOS theory concept educational mode provides some room for students to research by themselves, find the problems and solve them. It’s helpful to strengthen students’awareness of exploration and it demonstrates feasibility and effectiveness of the APOS theoretical mode.
According to research and analysis of implementing the APOS theory concept educational mode in practical teaching, this article finally puts forward several suggestions in implementing the APOS theory concept teaching process. First, the four phases of APOS theory is a relatively continuous process. Each phase is indispensable during the establishment of a concept in the brain. Second, Creating situations is not the ultimate goal of the concept teaching. Concept teaching shouldn’t stay on activities (operational) level, and take other stages hastily. Third, the establishment of mathematical concepts from the "process" to "targeted" needs to be repeated, step by step. Fourth, The object and thescheme stages should be stressed, and they can be recycled up. At last, the four stages are not necessarily reflected in a math class, nor every lesson must traverse the four stages. It applies to the establishment of a mathematical concept in the minds of students for a period of time, not limited to a certain class.
美国教育学家杜宾斯基(Ed Dubinsky)等人在数学教育研究实践中提出的APOS理论,对数学概念教学很有启示。该理论集中于对特定学习内容——数学概念学习过程的研究,提出学生学习数学概念要经过“活动”、“过程”、“对象”、“图式”四个阶段。APOS理论强调把概念寓于现实社会背景中,学生通过活动亲身经历、体验与现实的联系,从中经历完整的学习过程,用数学方法组织和建立概念。这样建立起来的概念具有丰富的内涵,其中包含着概念的现实原型、概念的抽象过程、思想方法和概念的形式化等,而达到建构数学概念的目的,这与新课标所提倡的学生主动建构是相一致的,顺应了当前数学教育的改革。
本研究在探讨和分析APOS理论的基础上,把APOS理论应用于圆锥曲线概念教学,进一步检验APOS理论概念教学模式应用于数学概念教学的实施效果。研究分为理论研究与教学实证研究两部分。理论研究为教学实证研究奠定了基础,教学实证研究也使得理论研究成果得到更好的诠释,两者相互促进,是我们研究的基本思路。在对国内外的相关文献进行综述的基础上,基于建构主义理论和图式理论的研究,我们对APOS理论的四个阶段进行剖析,探讨APOS理论概念教学设计和实施过程,为开展数学概念教学研究和实践研究提供理论支撑和教学策略;实证研究的对象为柳州市一中高二年级的两个平行班学生共131人,在将近一个学期的调查研究中,从实验的准备阶段到实验的全过程,通过问卷调查、访谈、课堂观察、测试分析等方式,运用统计分析的工具对相关数据进行分析,并结合课堂教学的实际,分析在日常教学中实施APOS理论概念教学的可行性、教学实施的困难和效果。重点是对实验班和对照班学生的数学成绩、数学兴趣、数学探究意识进行比较研究,以及对同一教学内容采取APOS理论概念教学模式和传统概念教学模式的教学过程的比较分析。
教学实证研究的结果较好的切合了理论研究,实证研究的结果表明:运用APOS理论概念教学模式进行数学概念教学激发了学生参与教学的心理机制,激发了学习的热情,提高学生学习数学的兴趣,加深他们对圆锥曲线概念的理解,数学成绩有所提高;APOS理论概念教学模式给学生提供了自主探究、发现问题和解决问题的空间,有助于学生探究意识的加强,初步验证了APOS理论概念教学模式实践的可行性和实施的有效性。
根据在实际教学中实施APOS理论概念教学模式的分析研究,本文最后提出在实施APOS理论概念教学过程中需注意的几点建议:APOS理论的四个阶段是一个相对连续的过程,概念在大脑中建立期间任一阶段都是不可缺少的;创设情境并不是概念教学的最终目的,概念教学不能只停留在活动(操作)层面,而对其他阶段草草收场;数学概念由“过程”到“对象”的建立需经多次反复,循序渐进;对象、图式阶段是数学概念在头脑中建立的长远之计,二者可循环上升;APOS理论的四个阶段并非一定体现在一堂数学课当中,也不是每一课都必须遍历四个阶段,它适用于数学概念在学生头脑中建立的一段时期,并不局限于某一堂课。
Mathematical concept is the logical beginning of maths, the basis for students’cognition. It has an important position in mathematics study and the teaching. Since long ago, mathematics concepts teaching have usually been suggested in the way of Concept Form and Concept Assimilation. Most teachers adopt Concept Assimilation. This kind of teaching process is relatively simple and enable students to learn concepts more directly, and it is called " themost essential way for students to obtain mathematical concepts ". However, this teaching method emphases on teaching the logical structure of concepts. A quite universal tendency is not to take the process of teaching and knowledge seriously.Then various types of exercises are introduced by teachers and students have to do a large number of imitation and practice. Students are unable to utilize the concepts which they obtain through the classroom instruction actually isunable to perform to utilize, thus causing many average students and students who have difficulty in learning to make mistakes when solving problems because of misunderstanding the concepts. The traditional teaching method of the mathematical concept has led to a series of negative consequences. It runs counter to " to the development of students" , the basic idea of elementary education put forward by our country . It’s advocated in the current curriculum standard that mathematical concepts teaching must be reverted to simplicity, reveals the formation of mathematical concepts, thus enabling students to multidimensionally understand the concepts from the prototype to abstract process, the guiding role of mathematical thinking, the use of symbols and forms of expression, which, conforms to the education principle of constructing the construction on their own initiative. Therefore, in order to promote the development of students, we must shift the concept teaching from the traditional model.
American education experts Grosso (Ed Dubinsky) and so on, proposed APOS Theory in mathematics education research practice, which is very enlightening to concept teaching. The theory focuses on the specific contents, that is the learning process of the mathematical concept and proposes four stages to learn mathematics concepts, that is“activity", "process", "object", and "scheme". After the students learn mathematical concepts to "activities" and "process" and "targets", "plans" four stages. APOS theory stressed that the concept resides in real social backgrounds. Students experience its links with reality through activities, experience the integrity of the study process, and use methodology to establish the concept. Concepts based on this method contains a rich connotation, which contains the concept’s prototype, abstract process, thinking method and formalization and so on. The goal of constrcting the mathematical concepts is achieved. It is consistent with“students taking the initiative”advocated in the New Curriculum Standard complies with the current mathematics education reform.
Based on discussing and analyzing the APOS theory, the research applies the APOS theory to the conic curve concept teaching and further examines the implementation effects of the APOS theory concept educational mode being applied to tomathematics concept teaching. The research consists of the fundamental research and teaching empirical study. The fundamental research lays the foundation for teaching empirical study, and the latter also makes a better interpretation for the former. They support each other, which is our basic idea. On the basis of the related literatures, and based on the schema theory of constructivism theory, we analyse the four-phases of APOS theory, and explore the APOS theory concept teaching design and the implementation process, so as to provide theoretical support and teaching strategies for the development of the mathematics concept teaching research and the practical research. Empirical research covers 131 Senior 2 students in two classes in Liuzhou No 1 Middle School. In six months of research, from the preparation to the entire process, through questionnaire surveys, interviews, classroom observation, test analysis, analyze the data by using statistical analysis as a tool, combine with the actual classroom teaching, analyse feasibility and effectiveness or difficulty in implementing APOS theoretical concepts in the regular teaching. The key point is to compare students’mathematics results, mathematics interest, mathematical awareness of exploration between key class and the tested classes, as well as compare and analyse the teaching processes on the same teaching content between the two teaching modes, the APOS theory concepteducational mode and the traditional concept educational mode.
Empirical research findings better fits the theoretical study. The results show that math concept teaching based on inspires students to participate in teaching and learning, stimulate students’interest in mathematics, enhance their understanding of the concept of conic curve, and do better in math. The APOS theory concept educational mode provides some room for students to research by themselves, find the problems and solve them. It’s helpful to strengthen students’awareness of exploration and it demonstrates feasibility and effectiveness of the APOS theoretical mode.
According to research and analysis of implementing the APOS theory concept educational mode in practical teaching, this article finally puts forward several suggestions in implementing the APOS theory concept teaching process. First, the four phases of APOS theory is a relatively continuous process. Each phase is indispensable during the establishment of a concept in the brain. Second, Creating situations is not the ultimate goal of the concept teaching. Concept teaching shouldn’t stay on activities (operational) level, and take other stages hastily. Third, the establishment of mathematical concepts from the "process" to "targeted" needs to be repeated, step by step. Fourth, The object and thescheme stages should be stressed, and they can be recycled up. At last, the four stages are not necessarily reflected in a math class, nor every lesson must traverse the four stages. It applies to the establishment of a mathematical concept in the minds of students for a period of time, not limited to a certain class.
引文
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