“本原性数学问题驱动课堂教学”的比较研究
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摘要
本论文是关于改进数学教师课堂教学、促进其专业发展的校本的行动研究。笔者从对数学本身的认识出发、结合我国当前的数学教学现状、响应数学和数学教育专家“注重实质”的倡导，提出一种“本原性数学问题驱动课堂教学”的理念。它持“动态的拟经验主义”数学观，提倡数学教学应扎根于学生的常识和经验，超越对数学技巧性的过度追求、深入到情境性问题的数学核心；让学生经历类似数学家的数学活动过程——数学的猜想、合情推理、(试误)探究、检验、证明等，并不断重组新的常识或经验，学习所教主题的数学本质。
研究的对象是两位小学教师“有余数除法”的教学和两位初中教师“勾股定理”的教学，这里的“两位”分别是职初教师、经验教师。要求每位教师就同一主题进行三轮授课：常规的自然状态下的教学；尝试实施“本原性数学问题驱动的课堂教学”和改进实施“本原性数学问题驱动的课堂教学”。研究的问题是：在“本原性数学问题驱动课堂教学”这一教学理念指导下，职初与经验教师实施的课堂教学有何差异、改变和共同之处?
研究过程中，质的研究是基本方法。从第一轮课后开始，教师和同事、研究者们开展集体备课、评课、研讨等教研活动。笔者始终参与其中，采用了课堂录像、课后访谈、反思日记、记录教研活动等手段收集资料。分析主要采用了匹兹堡大学“QUASAR课题”研究成果中的“数学任务框架”、“任务分析指南”、“保持和降低高认知要求任务的因素”等作为工具，并使用了录像带分析技术和语言交流的语义分析。研究的结果以横向比较和纵向比较的方式呈现。横向比较针对职初和经验教师的每一轮教学的异同，纵向比较则针对每位教师的前后三轮教学的变化。
通过横向比较，发现每轮授课各有三个共同点。第一轮授课：(1) 具有一定的“套路”、程式化的教学模式；(2) 强调知识结果的获得，较少关注数学过程，忽视本质；(3) 教师对经验的依赖程度偏高却对教学设计缺乏理性的反思。第二轮授课：(1) 教学背上了“理念的包袱”，授课内容繁杂且严重超时；(2) 强调了知识的发生过程，关注到学生的数学活动过程；(3) 教师过于依赖集体备课的
摘要
的设计思想。第三轮授课:(l)表现出相同或相似的教学过程，教学环节融合、
表现出连贯性;(2)重点关注学生的学习获得;(3)教学表现为集体智慧的优势。
纵向比较中发现两个重要现象:职初和经验教师的三轮授课经历了“关注教
材和个人经验”、“关注新理念和他人经验”到“关注现实课堂中学生的获得”的
变化过程;各轮授课进行中，职初与经验教师的授课环节与时间分配逐渐出现令
人惊讶的相似或相同。
通过“本原性数学问题驱动课堂教学”理念指导，教师的共同点表现在:(l)
教师对授课主题的思想有了深刻理解;(2)教师的教学行为发生了改变，数学知
识发生过程被凸现出来、为维持学生高认知水平的行为增加了;(3)教师行为的
改变经历了一个“痛苦”的过程:(4)教师的教育理论观和反思意识增强了。
职初与经验教师也表现出了差异:(:1)在实施常规性学习任务时经验教师优
于职初教师;而在实施挑战性学习任务时则反之。(2)在教学观念层面，经验教
师的教材观明显发生转变，学生观较为丰富和深刻;而职初教师在这两方面表现
不够明显。(3)反思的深刻性方面，职初教师不如经验教师，也不象他们那样倾
向于从一种整体、宏观的角度来表达个人的观点。
通过本行动研究，笔者对‘本原性数学问题驱动课堂教学”认识深化了。它
是一个围绕什么是所教数学主题中“最为本质的、基本的要素或构成”这个基本
问题扩展了的问题系统，是一个回归课堂教学实践的动态思考过程。它不仅指实
施的或理想的“本原性数学问题驱动的课堂教学”，而且指教师带着“本原性数
学问题驱动课堂教学”的“意识”，在实践中不断调整教学、符合课堂现实的一
种“思维方式气
根据研究结论，笔者提出了一些教师教育方面的建议:研究人员介入的教研
活动是教师教育中“不连续性教育”的一条途径:教师通过行动后反思要学会“向
自己学习”;反思意识的培养要有具体的“反思内容”;教师观念与行为的同步改
进，·关键是要存在“认知冲突”;“本原性数学问题驱动课堂教学”作为一种思考
方式，可以成为教师批判性“再思考理念本身”的工具。
The dissertation is a result of an action research on improving mathematics teachers' classroom teaching and facilitating their professional development. The author bring forward a teaching belief which is based on the understanding of mathematics, the state quo of mathematics teaching in China and the emphasis on "mathematics essence" by some experts in mathematics or mathematics education field, called "Classroom Teaching Driven by Primitive Mathematics Idea"(abbreviated as CTDPMI). It is guided by quasi-empiricism mathematics belief, call for that mathematics teaching should root in students' common sense and experience, surpass too much pursuit of skilled mathematics and embed the core of situated mathematics; let students master the essence of mathematics what taught by teachers, through the quasi-mathematician activities--mathematical supposing, rational reasoning, (error-trial) exploring, validating, proving et al, and reorganize students' new common sense and experience.
The research objects are two primary school teachers' teaching of "Division with a complement" and two junior high school teachers' teaching of "Pythagorean theorem", here the "two" means mat one is a novice and the other is an experienced teacher. Each teacher was asked to teach the same theme for three times. The first time of teaching is generally a natural one; the second, attempting CTDPMI, and the third, improving CTDPMI. The research question is: What's the difference, change or similarity between novice and experienced teachers in their practice of CTDPMI?
In the study, qualitative research is the basic way. From the end of the first natural teaching, four teachers, their colleagues and researchers prepared, reviewed and discussed the lesson all together. The author attended the teaching research activities all the while, and collected data by video-taped classroom, interviewing each teacher after her lesson, reflective diary and audio-taped teaching research activities. Some tools were used to analyze the collected data: The Mathematics Tasks Framework, Guide For Tasks Analysis and Factors Associated With Maintenance And Decline Of High-level Cognitive Demands in QUASAR project, Pittsburgh University, some techniques of video-taped analysis and discourse semantic analysis. The researching results were presented in
East China Normal University
transverse and longitudinal way. The transverse comparison aimed at the difference and similarity between novice and experienced teachers in each time of teaching, and the longitudinal comparison aimed at the change of each teacher m her three times of teaching.
In the transverse comparison, three common grounds were found in each time of teaching. In the first time of teaching, (1) mere was a certain routine way, procedurized pattern of teaching; (2) the product of knowledge was emphasized and process of mathematics was neglected, and yet essence; (3) teachers highly depended on their experiences but short of reflection on their teaching design. In the second time of teaching, (1) it seemed teaching was given a "belief burden"; (2) but the process of knowledge and students' mathematical activities were emphasized; (3) teachers depended on the collective ideas about teaching too much. In the third time of teaching, (1) they had same or similar teaching process, and the taches of teaching were integrated and consistent; (2) the students' acquirement was stressed; (3) the teaching was a result of the collective wits.
In the longitudinal comparison, two signified phenomena were found: all the teachers' three times of teaching had the same changing
process--from the beginning "paying attention to textbook and personal
experience", to the next "paying more attention on new belief and others experience", and to last "paying high attention on acquirement of the students in the realized classroom". The second phenomenon was that novice and experienced teachers little by little have the surprising similarity or common ground in teaching taches and time distribution in the whole process of the t
研究的对象是两位小学教师“有余数除法”的教学和两位初中教师“勾股定理”的教学，这里的“两位”分别是职初教师、经验教师。要求每位教师就同一主题进行三轮授课：常规的自然状态下的教学；尝试实施“本原性数学问题驱动的课堂教学”和改进实施“本原性数学问题驱动的课堂教学”。研究的问题是：在“本原性数学问题驱动课堂教学”这一教学理念指导下，职初与经验教师实施的课堂教学有何差异、改变和共同之处?
研究过程中，质的研究是基本方法。从第一轮课后开始，教师和同事、研究者们开展集体备课、评课、研讨等教研活动。笔者始终参与其中，采用了课堂录像、课后访谈、反思日记、记录教研活动等手段收集资料。分析主要采用了匹兹堡大学“QUASAR课题”研究成果中的“数学任务框架”、“任务分析指南”、“保持和降低高认知要求任务的因素”等作为工具，并使用了录像带分析技术和语言交流的语义分析。研究的结果以横向比较和纵向比较的方式呈现。横向比较针对职初和经验教师的每一轮教学的异同，纵向比较则针对每位教师的前后三轮教学的变化。
通过横向比较，发现每轮授课各有三个共同点。第一轮授课：(1) 具有一定的“套路”、程式化的教学模式；(2) 强调知识结果的获得，较少关注数学过程，忽视本质；(3) 教师对经验的依赖程度偏高却对教学设计缺乏理性的反思。第二轮授课：(1) 教学背上了“理念的包袱”，授课内容繁杂且严重超时；(2) 强调了知识的发生过程，关注到学生的数学活动过程；(3) 教师过于依赖集体备课的
摘要
的设计思想。第三轮授课:(l)表现出相同或相似的教学过程，教学环节融合、
表现出连贯性;(2)重点关注学生的学习获得;(3)教学表现为集体智慧的优势。
纵向比较中发现两个重要现象:职初和经验教师的三轮授课经历了“关注教
材和个人经验”、“关注新理念和他人经验”到“关注现实课堂中学生的获得”的
变化过程;各轮授课进行中，职初与经验教师的授课环节与时间分配逐渐出现令
人惊讶的相似或相同。
通过“本原性数学问题驱动课堂教学”理念指导，教师的共同点表现在:(l)
教师对授课主题的思想有了深刻理解;(2)教师的教学行为发生了改变，数学知
识发生过程被凸现出来、为维持学生高认知水平的行为增加了;(3)教师行为的
改变经历了一个“痛苦”的过程:(4)教师的教育理论观和反思意识增强了。
职初与经验教师也表现出了差异:(:1)在实施常规性学习任务时经验教师优
于职初教师;而在实施挑战性学习任务时则反之。(2)在教学观念层面，经验教
师的教材观明显发生转变，学生观较为丰富和深刻;而职初教师在这两方面表现
不够明显。(3)反思的深刻性方面，职初教师不如经验教师，也不象他们那样倾
向于从一种整体、宏观的角度来表达个人的观点。
通过本行动研究，笔者对‘本原性数学问题驱动课堂教学”认识深化了。它
是一个围绕什么是所教数学主题中“最为本质的、基本的要素或构成”这个基本
问题扩展了的问题系统，是一个回归课堂教学实践的动态思考过程。它不仅指实
施的或理想的“本原性数学问题驱动的课堂教学”，而且指教师带着“本原性数
学问题驱动课堂教学”的“意识”，在实践中不断调整教学、符合课堂现实的一
种“思维方式气
根据研究结论，笔者提出了一些教师教育方面的建议:研究人员介入的教研
活动是教师教育中“不连续性教育”的一条途径:教师通过行动后反思要学会“向
自己学习”;反思意识的培养要有具体的“反思内容”;教师观念与行为的同步改
进，·关键是要存在“认知冲突”;“本原性数学问题驱动课堂教学”作为一种思考
方式，可以成为教师批判性“再思考理念本身”的工具。
The dissertation is a result of an action research on improving mathematics teachers' classroom teaching and facilitating their professional development. The author bring forward a teaching belief which is based on the understanding of mathematics, the state quo of mathematics teaching in China and the emphasis on "mathematics essence" by some experts in mathematics or mathematics education field, called "Classroom Teaching Driven by Primitive Mathematics Idea"(abbreviated as CTDPMI). It is guided by quasi-empiricism mathematics belief, call for that mathematics teaching should root in students' common sense and experience, surpass too much pursuit of skilled mathematics and embed the core of situated mathematics; let students master the essence of mathematics what taught by teachers, through the quasi-mathematician activities--mathematical supposing, rational reasoning, (error-trial) exploring, validating, proving et al, and reorganize students' new common sense and experience.
The research objects are two primary school teachers' teaching of "Division with a complement" and two junior high school teachers' teaching of "Pythagorean theorem", here the "two" means mat one is a novice and the other is an experienced teacher. Each teacher was asked to teach the same theme for three times. The first time of teaching is generally a natural one; the second, attempting CTDPMI, and the third, improving CTDPMI. The research question is: What's the difference, change or similarity between novice and experienced teachers in their practice of CTDPMI?
In the study, qualitative research is the basic way. From the end of the first natural teaching, four teachers, their colleagues and researchers prepared, reviewed and discussed the lesson all together. The author attended the teaching research activities all the while, and collected data by video-taped classroom, interviewing each teacher after her lesson, reflective diary and audio-taped teaching research activities. Some tools were used to analyze the collected data: The Mathematics Tasks Framework, Guide For Tasks Analysis and Factors Associated With Maintenance And Decline Of High-level Cognitive Demands in QUASAR project, Pittsburgh University, some techniques of video-taped analysis and discourse semantic analysis. The researching results were presented in
East China Normal University
transverse and longitudinal way. The transverse comparison aimed at the difference and similarity between novice and experienced teachers in each time of teaching, and the longitudinal comparison aimed at the change of each teacher m her three times of teaching.
In the transverse comparison, three common grounds were found in each time of teaching. In the first time of teaching, (1) mere was a certain routine way, procedurized pattern of teaching; (2) the product of knowledge was emphasized and process of mathematics was neglected, and yet essence; (3) teachers highly depended on their experiences but short of reflection on their teaching design. In the second time of teaching, (1) it seemed teaching was given a "belief burden"; (2) but the process of knowledge and students' mathematical activities were emphasized; (3) teachers depended on the collective ideas about teaching too much. In the third time of teaching, (1) they had same or similar teaching process, and the taches of teaching were integrated and consistent; (2) the students' acquirement was stressed; (3) the teaching was a result of the collective wits.
In the longitudinal comparison, two signified phenomena were found: all the teachers' three times of teaching had the same changing
process--from the beginning "paying attention to textbook and personal
experience", to the next "paying more attention on new belief and others experience", and to last "paying high attention on acquirement of the students in the realized classroom". The second phenomenon was that novice and experienced teachers little by little have the surprising similarity or common ground in teaching taches and time distribution in the whole process of the t
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