初中数学教师错误分析能力研究
|
摘要
随着教育改革的不断推进，教师教育和教师专业化得到了广泛的重视和加强，教师的专业能力越来越成为教师专业发展的重中之重，因而对于教师专业能力的研究显得尤为迫切。对学生的学习错误进行分析是教师的一项基本教学活动，错误分析能力是教师从事教学工作必需具备的一种重要的专业能力。但已有的研究多集中在错误分析活动本身，研究者并没有从教师能力的视角入手，对教师在错误分析这一特殊活动中所表现出来的能力展开深入系统的研究。由于缺乏错误分析能力这一明确的核心概念，相关的研究不仅零散，而且应用价值受到了限制。
本研究以数学活动的教学理论和认知建构理论为基础，提出了数学教师错误分析能力的学科概念。所谓数学教师错误分析能力，是指数学教师为了保证学生数学学习的高效和成功，将学生学习中的数学错误作为思维对象，对其进行识别、解释、评估和纠正的能力。在此概念基础之上，研究了初中数学教师错误分析能力的问题，具体来说，主要研究了三个问题：初中数学教师错误分析能力的现状如何；影响初中数学教师错误分析能力的因素有哪些；初中数学教师错误分析能力发展的主要途径和策略是什么。
基于研究问题的探索性与开创性，整个研究以现象学的相关理论作为研究的方法论基础，采用质的研究方法进行数据的收集与分析。具体的方法为问卷法和访谈法。问卷由3份具有同样效度的问卷组成，每份问卷的核心部分是4个关于初中学生的数学错误及研究者所设置的相应问题，问卷对象涉及107位初中数学教师。访谈包括基于任务的访谈和半结构性的深度访谈。基于任务的访谈主要是在问卷的基础上展开的，对象为来自问卷中的36位教师，其中有18位教师包括9位专家型教师和9位普通教师接受了二次访谈。半结构性的深度访谈主要针对9位专家型教师错误分析能力发展的途径而展开。
通过研究，得到以下结论：
(一)初中数学教师错误分析能力现状的研究结论
初中数学教师错误分析能力存在4种不同层次的水平：零分析水平，模糊分析水平，部分-准确分析水平和全面-准确分析水平，它们反映了初中数学教师错误分析能力的某种内在的规律特征。从当前的现实情况来看，初中数学教师错误分析能力水平整体偏低，多数处于模糊分析水平。
初中数学教师错误分析能力存在的问题是：(1)初中数学教师难以识别逻辑性错误。(2)对于学生的数学错误，初中数学教师普遍存在“知其然而不知其所以然”的现象。(3)初中数学教师多数从学生的角度归因数学错误。
初中数学教师错误分析能力有三种不同的表现：(1)“自觉性”与“不自觉性”；(2)“合教师性”与“合学生性”；(3)“开放性”与“封闭性”。
(二)影响初中数学教师错误分析能力因素的研究结论
教师知识和教师观念是影响初中数学教师错误分析能力的两个主要因素。
专家型教师与普通教师在关于学科知识、教学知识、学生知识和课程知识方面的区别主要在于知识的多或少、丰富或贫乏；专家型教师与普通教师在关于数学观、教学观、学习观、错误观和自我效能感这几个方面的区别主要在于观念是否坚定明确，教师是否有自己的“理论”。
专家型教师和普通教师在知识和观念上的不同特点，影响他们在错误识别、错误解释、错误评估和错误纠正时的不同决策，从而影响他们错误分析能力的高低。因而提高初中数学教师错误分析能力就必须在改善教师知识和教师观念这两方面的素质上下功夫。
(三)初中数学教师错误分析能力发展的途径和策略的研究结论
初中数学教师错误分析能力发展的途径主要来源于教师教学实践中的反思，它包括经验性反思、交流性反思和学习性反思三个方面。正规的职前教师教育与职后的专业活动并不是初中数学教师错误分析能力发展的主要途径。
初中数学教师错误分析能力的发展可以从两方面着手。一种是从教师自身入手，促使初中数学教师学会在教学实践中批判性反思。另一种是从外部入手，对初中数学教师错误分析能力的发展进行培训干预，即在传授一定的关于错误分析的理论知识的基础上，以学生的数学错误的案例分析作为重点，同时辅之以其它数学任务方面的内容。
基于以上的研究发现和结论，本研究针对教师教育工作者和数学教师得到了一些关于如何培养数学教师错误分析能力的启示，也探讨了一些值得进一步研究的问题。
本研究的创新之处在于：
(1)在数学教师的错误分析活动和教师能力这两者的交叉研究领域内作了初步的尝试，首次将数学教师错误分析能力作为一个独立的研究对象，对初中数学教师错误分析能力的现状、影响因素、发展途径和策略进行了较为系统深入的研究。
(2)提出了数学教师错误分析能力结构模型的构想，为科学解释数学教师在错误分析这一特殊活动中的能力作出了初步的尝试。
(3)在研究方法上没有按照传统的依靠定量数据的统计分析来研究能力，而是采用质的研究方法，通过分析教师所表现的质的特性来推断其错误分析的思维过程。
With the education reform goes depth, teacher's education and professionalization get more and more attentions and strengthening. The teacher's professional ability growingly becomes the most important one in the field of teacher's professional development. As a consequence, it is imperative to study teacher's professional ability and its development. Due to that analyzing students' errors in learning is an every-day work and basic activity, the ability of error analysis is an essential and indispensable professional ability for a teacher to conduct teaching. However, the most of existed researches didn't systematically and deeply study this ability represented in the special activity of error analysis from the point of view of teacher's professional ability while focused only on the error analysis itself. For the lack of the core conception of the ability of error analysis, the existed relevant researches are scattered and their application value are also limited.
Based on the teaching theory of mathematical activity and the cognitive constructive theory, the subject conception of mathematics teacher's ability of error analysis is presented. In this dissertation, the mathematics teacher's ability of error analysis is defined as the mathematics teacher's ability to identify, to explain, to evaluate and to correct students' mathematical errors in mathemtics learning, in which the mathematics teacher take students' mathematical errors as his or her thought object in order to ensure students' high efficiency and success in mathematics learning. Based on this definition, the mathematics teacher's ability of error analysis is studied. Speaking concretely, three main questions are studied in this dissertation: what is the current status of the ability of error analysis of mathematics teachers of junior high school, what are the factors influencing the ability of error analysis of mathematics teachers of junior high school and what are the leading avenues and strategies for developing the ability of error analysis of mathematics teachers of junior high school.
Due to its exploration and initiation, the methodology of this dissertation is based on the phenomenological theory, and the data are collected and analyzed through the qualitative method which has widespread use in social science. The concrete approaches are questionnaires and interviews. The questionnaire consists of three questionnaires which have the same efficiency. The core components of each questionnaire are four contexts and problems relevant to the mathematical errors of students of junior high school. 107 mathematics teachers of junior high school were involved in the questionnaire investigating. The interviews include two types: based on the tasks and based on the semi-structure. The tasks-based interviews are based on the questionnaire investigating results. Its object is to understand the teachers' thinking processes more detailedly and clearly, in which the special attentions are particularly put on the underlying reasons for the mathematics teachers' decision-making during error analysis. 36 teachers of the 107 mathematics teachers involved in the questionnaire investigating were interviewed, among them 18 teachers were interviewed at second time consisting of 9 ordinary teachers and 9 expert teachers. The semi-structure-based interview is to directly or indirectly find out the avenues and strategies for developing teachers' ability of error analysis, in which 9 expert teachers were involved.
Through studying, we got many findings and conclusions as follows. Firstly, the conclusion with respect to the current status of the ability of error analysis of mathematics teachers of junior high school is that there are four different levels in the ability of error analysis of mathematics teachers of junior high school.
The four different levels are 'goose egg' analysis level, indistinct analysis level, partly exact analysis level and totally exact analysis level respectively, which reflect certain internal law's characteristics of the ability of error analysis of mathematics teachers of junior high school. From the point of view of current reality status, as a whole, the ability of error analysis of the mathematics teachers of junior high school is low; most of them are in the position of indistinct analysis level.
There are many problems in the ability of error analysis of the mathematics teachers of junior high school, namely, (1) the identifying of the students' logic errors in mathematics learning is difficult for mathematics teachers of junior high school, (2) there widely exists 'knowing that without knowing why' phenomenon to students' mathematical errors in mathematics teachers of junior high school, (3) a majority of mathematics teachers of junior high school attribute students' errors to students themselves.
The ability of error analysis of the mathematics teachers of junior high school shows three different characteristics: the 'consciousness' and the 'sub-consciousness', the 'pertaining to student' and the 'pertaining to teacher', and the 'openness' and the 'closeness'.
Secondly, the conclusion about the factors influencing the ability of error analysis of the mathematics teachers of junior high school is that teacher's knowledge and ideas are the main factors influencing the ability of error analysis of mathematics teachers of junior high school.
As far as the knowledge about the subject matter, the teaching, the curriculum and students is concerned, the difference between ordinary teachers and expert teachers mainly lies in the enrichment or shortage of relevant knowledge. Meanwhile, in mathematics belief, teaching belief, learning belief, error belief and self-confidence, the difference between ordinary teachers and expert teachers mainly lies in that whether their ideas are firm and definite and whether they have their own 'theories'.
The different behaviors on knowledge and belief between ordinary teachers and expert teachers lead to different decision-makings in the identification, explanation, evaluation and correction of students' mathematical errors, and then lead to different levels of ability of error analysis. Thus, in order to develop the ability of error analysis of the mathematics teachers of junior high school, the efforts to be made on the improvement of their knowledge and belief.
Thirdly, as for the avenues and strategies for developing the ability of error analysis of mathematics teachers of junior high school, the study conclusion is that the development of the ability of error analysis of mathematics teachers of junior high school mainly comes from teacher's reflection in teaching practice, which includes the reflection from experiences, the reflection from communication and the reflection from study. The formal pre-service education and the in-service professional activities don't fall into the main avenues and strategies for developing the ability of error analysis of mathematics teachers of junior high school.
Two strategies are planned for the development of the ability of error analysis of mathematics teachers of junior high school. The first one is that starting from the teachers themselves, to prompt the mathematics teachers of junior high school to learn to reflect skeptically. The second one is starting from the external of teachers, to improve the development of the ability of error analysis of mathematics teachers of junior high school, i.e., based on imparting certain theoretical knowledge about the error analysis to mathematics teachers of junior high school, taking cases analysis of students' typical mathematical errors as the emphasis, to impart contents of other mathematical tasks to them.
Based on the above findings and conclusions, some suggestions about the cultivating of the ability of error analysis of mathematics teachers both for the mathematics teacher educators and the mathematics teachers are presented, furthermore, some issues to be further studied are also explored.
The innovation points of this dissertation are as follows: (1) the ability of error analysis of mathematics teachers of junior high school is firstly studied relatively systematically and deeply in home, (2) the structural model of the ability of error analysis of mathematics teachers is presented, which makes a primary attempt to scientifically explain the ability of mathematics teachers presented in the special activity of error analysis and (3) this research is conducted through a complete procedure of qualitative methodology rather than the traditional quantitative methodology. The thinking processes of teachers in error analysis are extrapolated by analyzing teachers' essential characteristics showed in the activity of error analysis.
本研究以数学活动的教学理论和认知建构理论为基础，提出了数学教师错误分析能力的学科概念。所谓数学教师错误分析能力，是指数学教师为了保证学生数学学习的高效和成功，将学生学习中的数学错误作为思维对象，对其进行识别、解释、评估和纠正的能力。在此概念基础之上，研究了初中数学教师错误分析能力的问题，具体来说，主要研究了三个问题：初中数学教师错误分析能力的现状如何；影响初中数学教师错误分析能力的因素有哪些；初中数学教师错误分析能力发展的主要途径和策略是什么。
基于研究问题的探索性与开创性，整个研究以现象学的相关理论作为研究的方法论基础，采用质的研究方法进行数据的收集与分析。具体的方法为问卷法和访谈法。问卷由3份具有同样效度的问卷组成，每份问卷的核心部分是4个关于初中学生的数学错误及研究者所设置的相应问题，问卷对象涉及107位初中数学教师。访谈包括基于任务的访谈和半结构性的深度访谈。基于任务的访谈主要是在问卷的基础上展开的，对象为来自问卷中的36位教师，其中有18位教师包括9位专家型教师和9位普通教师接受了二次访谈。半结构性的深度访谈主要针对9位专家型教师错误分析能力发展的途径而展开。
通过研究，得到以下结论：
(一)初中数学教师错误分析能力现状的研究结论
初中数学教师错误分析能力存在4种不同层次的水平：零分析水平，模糊分析水平，部分-准确分析水平和全面-准确分析水平，它们反映了初中数学教师错误分析能力的某种内在的规律特征。从当前的现实情况来看，初中数学教师错误分析能力水平整体偏低，多数处于模糊分析水平。
初中数学教师错误分析能力存在的问题是：(1)初中数学教师难以识别逻辑性错误。(2)对于学生的数学错误，初中数学教师普遍存在“知其然而不知其所以然”的现象。(3)初中数学教师多数从学生的角度归因数学错误。
初中数学教师错误分析能力有三种不同的表现：(1)“自觉性”与“不自觉性”；(2)“合教师性”与“合学生性”；(3)“开放性”与“封闭性”。
(二)影响初中数学教师错误分析能力因素的研究结论
教师知识和教师观念是影响初中数学教师错误分析能力的两个主要因素。
专家型教师与普通教师在关于学科知识、教学知识、学生知识和课程知识方面的区别主要在于知识的多或少、丰富或贫乏；专家型教师与普通教师在关于数学观、教学观、学习观、错误观和自我效能感这几个方面的区别主要在于观念是否坚定明确，教师是否有自己的“理论”。
专家型教师和普通教师在知识和观念上的不同特点，影响他们在错误识别、错误解释、错误评估和错误纠正时的不同决策，从而影响他们错误分析能力的高低。因而提高初中数学教师错误分析能力就必须在改善教师知识和教师观念这两方面的素质上下功夫。
(三)初中数学教师错误分析能力发展的途径和策略的研究结论
初中数学教师错误分析能力发展的途径主要来源于教师教学实践中的反思，它包括经验性反思、交流性反思和学习性反思三个方面。正规的职前教师教育与职后的专业活动并不是初中数学教师错误分析能力发展的主要途径。
初中数学教师错误分析能力的发展可以从两方面着手。一种是从教师自身入手，促使初中数学教师学会在教学实践中批判性反思。另一种是从外部入手，对初中数学教师错误分析能力的发展进行培训干预，即在传授一定的关于错误分析的理论知识的基础上，以学生的数学错误的案例分析作为重点，同时辅之以其它数学任务方面的内容。
基于以上的研究发现和结论，本研究针对教师教育工作者和数学教师得到了一些关于如何培养数学教师错误分析能力的启示，也探讨了一些值得进一步研究的问题。
本研究的创新之处在于：
(1)在数学教师的错误分析活动和教师能力这两者的交叉研究领域内作了初步的尝试，首次将数学教师错误分析能力作为一个独立的研究对象，对初中数学教师错误分析能力的现状、影响因素、发展途径和策略进行了较为系统深入的研究。
(2)提出了数学教师错误分析能力结构模型的构想，为科学解释数学教师在错误分析这一特殊活动中的能力作出了初步的尝试。
(3)在研究方法上没有按照传统的依靠定量数据的统计分析来研究能力，而是采用质的研究方法，通过分析教师所表现的质的特性来推断其错误分析的思维过程。
With the education reform goes depth, teacher's education and professionalization get more and more attentions and strengthening. The teacher's professional ability growingly becomes the most important one in the field of teacher's professional development. As a consequence, it is imperative to study teacher's professional ability and its development. Due to that analyzing students' errors in learning is an every-day work and basic activity, the ability of error analysis is an essential and indispensable professional ability for a teacher to conduct teaching. However, the most of existed researches didn't systematically and deeply study this ability represented in the special activity of error analysis from the point of view of teacher's professional ability while focused only on the error analysis itself. For the lack of the core conception of the ability of error analysis, the existed relevant researches are scattered and their application value are also limited.
Based on the teaching theory of mathematical activity and the cognitive constructive theory, the subject conception of mathematics teacher's ability of error analysis is presented. In this dissertation, the mathematics teacher's ability of error analysis is defined as the mathematics teacher's ability to identify, to explain, to evaluate and to correct students' mathematical errors in mathemtics learning, in which the mathematics teacher take students' mathematical errors as his or her thought object in order to ensure students' high efficiency and success in mathematics learning. Based on this definition, the mathematics teacher's ability of error analysis is studied. Speaking concretely, three main questions are studied in this dissertation: what is the current status of the ability of error analysis of mathematics teachers of junior high school, what are the factors influencing the ability of error analysis of mathematics teachers of junior high school and what are the leading avenues and strategies for developing the ability of error analysis of mathematics teachers of junior high school.
Due to its exploration and initiation, the methodology of this dissertation is based on the phenomenological theory, and the data are collected and analyzed through the qualitative method which has widespread use in social science. The concrete approaches are questionnaires and interviews. The questionnaire consists of three questionnaires which have the same efficiency. The core components of each questionnaire are four contexts and problems relevant to the mathematical errors of students of junior high school. 107 mathematics teachers of junior high school were involved in the questionnaire investigating. The interviews include two types: based on the tasks and based on the semi-structure. The tasks-based interviews are based on the questionnaire investigating results. Its object is to understand the teachers' thinking processes more detailedly and clearly, in which the special attentions are particularly put on the underlying reasons for the mathematics teachers' decision-making during error analysis. 36 teachers of the 107 mathematics teachers involved in the questionnaire investigating were interviewed, among them 18 teachers were interviewed at second time consisting of 9 ordinary teachers and 9 expert teachers. The semi-structure-based interview is to directly or indirectly find out the avenues and strategies for developing teachers' ability of error analysis, in which 9 expert teachers were involved.
Through studying, we got many findings and conclusions as follows. Firstly, the conclusion with respect to the current status of the ability of error analysis of mathematics teachers of junior high school is that there are four different levels in the ability of error analysis of mathematics teachers of junior high school.
The four different levels are 'goose egg' analysis level, indistinct analysis level, partly exact analysis level and totally exact analysis level respectively, which reflect certain internal law's characteristics of the ability of error analysis of mathematics teachers of junior high school. From the point of view of current reality status, as a whole, the ability of error analysis of the mathematics teachers of junior high school is low; most of them are in the position of indistinct analysis level.
There are many problems in the ability of error analysis of the mathematics teachers of junior high school, namely, (1) the identifying of the students' logic errors in mathematics learning is difficult for mathematics teachers of junior high school, (2) there widely exists 'knowing that without knowing why' phenomenon to students' mathematical errors in mathematics teachers of junior high school, (3) a majority of mathematics teachers of junior high school attribute students' errors to students themselves.
The ability of error analysis of the mathematics teachers of junior high school shows three different characteristics: the 'consciousness' and the 'sub-consciousness', the 'pertaining to student' and the 'pertaining to teacher', and the 'openness' and the 'closeness'.
Secondly, the conclusion about the factors influencing the ability of error analysis of the mathematics teachers of junior high school is that teacher's knowledge and ideas are the main factors influencing the ability of error analysis of mathematics teachers of junior high school.
As far as the knowledge about the subject matter, the teaching, the curriculum and students is concerned, the difference between ordinary teachers and expert teachers mainly lies in the enrichment or shortage of relevant knowledge. Meanwhile, in mathematics belief, teaching belief, learning belief, error belief and self-confidence, the difference between ordinary teachers and expert teachers mainly lies in that whether their ideas are firm and definite and whether they have their own 'theories'.
The different behaviors on knowledge and belief between ordinary teachers and expert teachers lead to different decision-makings in the identification, explanation, evaluation and correction of students' mathematical errors, and then lead to different levels of ability of error analysis. Thus, in order to develop the ability of error analysis of the mathematics teachers of junior high school, the efforts to be made on the improvement of their knowledge and belief.
Thirdly, as for the avenues and strategies for developing the ability of error analysis of mathematics teachers of junior high school, the study conclusion is that the development of the ability of error analysis of mathematics teachers of junior high school mainly comes from teacher's reflection in teaching practice, which includes the reflection from experiences, the reflection from communication and the reflection from study. The formal pre-service education and the in-service professional activities don't fall into the main avenues and strategies for developing the ability of error analysis of mathematics teachers of junior high school.
Two strategies are planned for the development of the ability of error analysis of mathematics teachers of junior high school. The first one is that starting from the teachers themselves, to prompt the mathematics teachers of junior high school to learn to reflect skeptically. The second one is starting from the external of teachers, to improve the development of the ability of error analysis of mathematics teachers of junior high school, i.e., based on imparting certain theoretical knowledge about the error analysis to mathematics teachers of junior high school, taking cases analysis of students' typical mathematical errors as the emphasis, to impart contents of other mathematical tasks to them.
Based on the above findings and conclusions, some suggestions about the cultivating of the ability of error analysis of mathematics teachers both for the mathematics teacher educators and the mathematics teachers are presented, furthermore, some issues to be further studied are also explored.
The innovation points of this dissertation are as follows: (1) the ability of error analysis of mathematics teachers of junior high school is firstly studied relatively systematically and deeply in home, (2) the structural model of the ability of error analysis of mathematics teachers is presented, which makes a primary attempt to scientifically explain the ability of mathematics teachers presented in the special activity of error analysis and (3) this research is conducted through a complete procedure of qualitative methodology rather than the traditional quantitative methodology. The thinking processes of teachers in error analysis are extrapolated by analyzing teachers' essential characteristics showed in the activity of error analysis.
引文
[1] 联合国教科文组织，赵中建主译．全球教育发展的历史轨迹：国际教育大会60年建议书．北京：教育科学出版社，1999：522．
[2] 戚业国，陈玉琨．学校发展与教师的专业发展．教育理论与实践，2002(8)：27-31．
[3] 联合国教科文组织总部中文科译．教育—财富蕴藏：其中国际21世纪教育委员会报告．北京：教育科学出版社，1996：14-15．
[4] 刘捷．专业化：挑战21世纪的教师．北京：教育科学出版社，2005：66-68．
[5] 范良火．教师知识发展研究．上海：华东师范大学出版社，2003：26-33．
[6] 教育部师范教育司．教师专业化的理论与实践．北京：人民教育出版社，2003：54．
[7] 李其龙，陈永明．教师教育课程的国际比较．北京：教育科学出版社，2002：139-140．
[8] 邵瑞珍．教育心理学—学与教的原理．上海：上海教育出版社，1983：265．
[9] 罗树华，李洪珍．教师能力学．济南：山东教育出版社，1997：27-73．
[10] Brown. H. D. Principles of Language Learning and Teaching. New Jersey: Prentice-Hall, 1987: 204-207.
[11] 魏红，申继亮．中小学教师评价学生能力的现状与对策．教育理论与实践，2004(5)：27-30．
[12] 申继亮，辛涛，邹泓．中小学教师教学能力观的比较研究．教育科学研究，1998(1)：1-4．
[13] 段宝斌，刘猛．对体育教师能力结构的认识．体育研究，2006(7)：163-164．
[14] Brown, J.S., & Burton, R.R. Diagnostic models for procedural bugs in basic mathematical skills. Cognitive Science, 1978(2): 155-192.
[15] 郑毓信，梁贯成．认知科学、建构主义与数学教育．上海：上海教育出版社，1998：62，229．
[16] 全美数学教师理事会著，蔡金法等译．美国数学教育的原则和标准．北京：人民教育出版社，2004：16-17，23-25．
[17] 王梓坤．今日数学及其应用．面向21世纪的数学教育—数学家论数学教育．南京：江苏教育出版社，1994：1-36．
[18] 郑君文，张恩华．数学学习论．南宁：广西教育出版社，2001：93．
[19] 张孝达．学生认为“(－3)×(－4)=9”该怎么办?中小学数学(教师版)，2004(5)：3．
[20] 李道路，孙朝仁．“有理数的乘法”教学实录及评析．中学数学杂志(初中)，2004(6)：15-19．
[21] 许芬英．从有理数乘法的引入看教材的多样化及其对教学的启示．中学数学教育，2004(10)．
[22] 曾飞鹏．关于“(－3)×(－4)=9”．中小学数学(教师版)，2004(7-8)：14-15．
[23] 梁国茂．关于学生认为“(－3)×(－4)=9”的解决办法．中小学数学(教师版)，2004(9)：9-10．
[24] 杨文鹏．读“(－3)×(－4)=9”有感．中小学数学(教师版)，2004(9)：10-11．
[25] 周金华，孙鹏国．关于“(－3)×(－4)=?”的教学．中小学数学(教师版)，2004(10)：6-7．
[26] 张成国．对一个典型案例的再思考．中小学数学(教师版)，2004(11)：7-9．
[27] 郑生．关于“有理数乘法”教学的探讨．中小学数学(教师版)，2004(12)：4-5．
[28] 李学华，申瑞青．读“有理数乘法教学”有感．中小学数学(教师版)，2005(3)：27-28．
[29] 李泽云．也谈“有理数乘法”的教学．中小学数学(教师版)，2005(4)：2-3．
[30] 章如芝．还谈“有理数乘法”的教学．中小学数学(教师版)，2005(7-8)：22-23．
[31] 汲广川．这样的情境不必设置．中小学数学(教师版)，2006(3)：25-26．
[32] 张广祥，张奠宙．代数教学中的模式直观．数学教育学报，2006(1)：1-4．
[33] 罗增儒．案例创作：“(－3)×(－4)=?”数轴表示的挑战．中学数学教学参考，2004(12)：5-7．
[34] 程来平．“1/2＋1/3＝2/5”和“1/2＋1/3＝5/6”具有同等价值．小学教学设计，2005(2)：52．
[35] 孙国元．“1/2＋1/3＝2/5”的价值在哪?小学教学设计，2005(7-8)：107．
[36] 蔡跟东．由“1/2＋1/3＝2/5”和“1/2＋1/3＝5/6”具有同等价值所想到的．小学教学设计，2005(7-8)：107．
[37] 郑毓信．数学教育哲学．成都：四川教育出版社(第二版)，2004：48-68，383，395，325．
[38] 李善良．现代认知观下的数学概念学习与教学．南京：江苏教育出版社，2005：286．
[39] 周友士．数学教学中要善待学生的错误．当代教育科学，2005(3)：52-54．
[40] Hiebert, J., & Carpenter, T.P. Learning and teaching with understanding. In Grouws, D.A. (eds.), Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics. New York: Macmillan, 1992: 65-97.
[41] Newton, D.P. Teaching for understanding. London and New York: Routledge Falmer, 2000.
[42] Ball, D.L. What do students know? Facing challenges of distance, context and desire in trying to hear children. In Biddle, B.J., Good, T.L., & Goodson, I.F. (eds.), International Handbook of Teachers and Teaching. Dordrecht, The Netherlands: Kluwer, 1997: 769-818.
[43] Wallach, T. Hearing students: the complexity of understanding what they are saying, showing, and doing. Journal of Mathematics Teacher Education, 2005(8): 393-417.
[44] Wildy, H., & Wallace, J. Understanding teaching or teaching for understanding. American Educational Research Journal, 1992: 29.
[45] Morgan, C., & Watson, A. The interpretative nature of teachers' assessment of students' mathematics: Issues for equity. Journal for Research in Mathematics Education, 2002(33): 78-110.
[46] Lin, P.J. Conceptualizing teachers' understanding of students' mathematical learning by using assessment tasks. International Journal of Science and Mathematics Education, 2006(4): 545-580.
[47] 朱德全，宋乃庆．论素质教育观下的数学教育．教育研究，1998(5)：23-27．
[48] 罗增儒．“以错纠错”的案例分析．中学数学教学参考，2001(9)：23-26．
[49] ICMI. The fifteenth ICMI study: the professional education and development of teachers of mathematics. Educational Studies in Mathematics, 2004(56): 359-372.
[50] 顾泠沅，易凌峰，聂必凯．寻找中间地带．上海：上海教育出版社，2003：54-164．
[51] Buswell, G.T., & Judd, C.H. Summary of Educational Investigations Relating to Arithmetic (Supplementary Educational Monograph No.27). Chicago: University of Chicago, 1925.
[52] Radatz, R. Error analysis in mathematics education. Journal for Research in Mathematics Education, 1979(5): 163-172.
[53] Mays, M. Misconceptions. In American Perspectives on the Seventh International Congress on Mathematical Education. Reston, Va.: National Council of Teachers of Mathematics, 1993.
[54] Langley, P., Wogulis, J., & Ohlsson, S. Rules and principles in cognitive diagnosis. In Frederiksen, N., Glaser, R., Lesgold, A.M., & Shafto, G. (eds.), Diagnostic Monitoring of Skill and Knowledge Acquisition. Hillsdale, NJ: Lawrence Erlbaum, 1990:217-249.
[55] Tatsuoka, K., & Corter, J. Patterns of diagnosed mathematical content and process skill in TIMSS-R across a sample of 20 countries. American Educational Research Journal, 2004, 41(4): 24-58.
[56] Sleeman, D., Kelly, A.E., Martinak, R., et al. Studies of diagnosis and remediation with high school algebra students. Cognitive Science, 1989(13): 551-568.
[57] Riccomini, P.J. Identification and remediation of systematic error patterns in subtraction. Learning Disability Quaterly, 2005(28): 233-242.
[58] David, S. On mathematical error. Studies in History and Philosophy of Science, 1997, 28(3): 393-415.
[59] Fiori, C., & Zuccheri, L. An experimental research on error patterns in written subtraction. Educational Studies in Mathematics, 2005(60): 323-331.
[60] 彭爱辉．国际数学教育委员会第二届克莱因奖和弗莱登塔尔奖得主一瞥．数学通报2006(11)：56-58．
[61] Bachelard, G. La Formation de l' Esprit Seientifique [The formation of scientific spirit]. Paris: Vrin, 1983.
[62] Brousseau, G. Les obstacles epistemologiques et les problemes en Mathematiques, [Epistemological obstacles and problems in mathematics], Recherches en Didactique des Mathematiques, 1983, 4(2): 164-198.
[63] Greer, B. The mathematics modeling perspective on world problems. Journal of mathematical behaviour, 1993(8): 143-160.
[64] Silver, E.A., Shapiro, L.J., & Deutch, A. Sense making and the solution of division problems involving remainders: an examination of middle school students' solutions processes and their interpretations of solutions. Journal for Research in Mathematics Education, 1993(24): 117-135.
[65] Cai, J., & Silver, E.A. Solution processes and interpretations of solutions in solving a division-with-remainder story problem: Do Chinese and U.S. students have similar difficulties? Journal for Research in Mathematics Education, 1995(26): 491-497.
[66] Brousseau, G. The crucial role of the didactical contract in the analysis and construction of situations in teaching and learning mathematics. In Steiner, H.G. (eds.), Theory of mathematics education. Occasional paper 54. Bielefeld: IDM, 1984: 110-119.
[67] Schoenfeld, A.H. Mathematical Problem Solving. New York: Academic Press, 1985.
[68] D'Amore, B., & Martini, B. Contrat didactiqque, modeles mentaux et modeles intutifs dans la resolution de problemes scolaires standard, [Students' responses to typical school problems with lack of data]. Scientia Paedagogica Experimentalis, 1998, 35(1): 55-94.
[69] Schoenfeld, A.H. What's all the fuss about metacognition? In Schoenfeld, A.H. (eds.), Cognitive Science and Mathematics Education. Hillsdale: Lawrence Erlbaum Ass, 1987: 189-215.
[70] Davis, R. Learning Mathematics: The Cognitive Approach to Mathematics Education. London: Routledge, 1989.
[71] Hald, M. E. A PASS cognitive processes intervention study in mathematics, 2000. http://wwwlib.global.umi.com/dissertations.
[72] Erlwanger, S.H. Benny's conception of rules and answers in IPI mathematics. The Journal of Children's Mathematical Behavior, 1973, 1(2): 7-26.
[73] 李士锜．PME：数学教育心理学．上海：华东师范大学出版社，2001：213-215．
[74] Bock, D.D., Verschaffel, L., & Janssens, D. The predominance of the linear model in secondary school students' solutions of word problems involving length and area of similar plane figures. Educational Studies in Mathematics, 1998(35): 65-83.
[75] Bock, D.D., Dooren, W., Janssens, D., et al. Improper use of linear reasoning: an in-depth study of the nature and the irresistibility of secondary school students' error. Educational Studies in Mathematics, 2002(50): 311-334.
[76] Fischbein, E. Intuition in Science and Mathematics. D.Reidel, Dordrecht. 1987.
[77] Economou, P. How teachers of mathematics confront students' errors. In Philippou, G., Christou, C., & Kakas, A. (eds.), Proceedings of the Second Panhellenic Conference on Mathematics Education and the Informatics in Education. Nicosia: Sighroni Epoxi, 1995: 383-400.
[78] Reason, J. Skill and Error in Everyday Life, Adult Learning: Psychological Research and Applications. New York: Wiley, 1977.
[79] [苏]舍瓦列夫，孙经灏译．学生作业中的概括联想．北京：科学出版社，1963：53．
[80] [苏]孜科娃，孙以芾译．掌握初等几何知识的心理概论．北京：人民教育出版社，1959：37，19-22．
[81] 卢仲衡．初二学生学习数学所产生的一些错误分析．数学通报，1961(7)：29-34．
[82] 卢仲衡，应玉叶，张梅玲等．关于如何减少图形交错的消极影响的问题．数学通报，1964(7)：16-19．
[83] [美]奥苏伯尔(Ausubel)，佘星南，宋钧译．教育心理学——认知观点．北京：人民教育出版社．1994：122-123．
[84] [苏]斯涅普坎，时勘等译．数学教学心理学．重庆：重庆出版社，1987：25．
[85] Tirosh, D., & Stavy, R. Intuitive rules: a way to explain and predict students' reasoning. Educational Studies in Mathematics, 1999(38): 51-69.
[86] Stavy, R. & Tirosh, D. How Students (Mis-)Understand Science and Mathematics: Intuitive Rules. Teachers College Press, 2000.
[87] Stavy, R., Babai, R., Tsamir, P., et al. Are intuitive rules universal? International Journal of Science and Mathematics Education, 2006(4): 417-436.
[88] Dooren, V., Bock, W. D., Weyers, D. D., et al. The predictive power of intuitive rules: A critical analysis of the impact of "more A-more B" and "same A-same B". Educational Studies in Mathematics, 2004(56): 179-207.
[89] Clement, J. Algebra word problem solutions: thought processes underlying a common misconception. Journal for Research in Mathematics Education, 1982, 13(1): 16—30.
[90] Leron, U., & Hazzan, O. The rationality debate: application of cognitive psychology to mathematics education. Educational Studies in Mathematics, 2006(62): 105-126.
[91] Chaiken, S., & Trope, Y. Dual-process Theories in Social Psychology. New York: Guilford, 1999.
[92] Ben, Z.T. The nature and origin of rational errors in arithmetic thinking: Introduction from examples and prior knowledge. Cognitive Science, 1995(19): 341-376.
[93] 吕林海．数学理解学习与教学研究．博士学位论文，华东师范大学，2005：99-109．
[94] Raffiella, B. Capitalizing on errors as "springboards for inquiry": a teaching experiment. Journal for Research in Mathematics Education, 1994, 25(2): 166-208.
[95] O'Connell, A. A. A classification of student errors in probability problem-solving. [Unpublished doctoral dissertation], Teachers College, Columbia University, New York, 1993.
[96] O'Connell, A.A. Understanding the nature of errors in probability problem-solving. Educational Research and Evaluation, 1999, 5(1): 1-21.
[97] 张奠宙主编．数学教育研究导引．南京：江苏教育出版社，1994：331-352．
[98] 张奠宙，宋乃庆．数学教育概论．北京：高等教育出版社，2005：7-8．
[99] Kieran，C．学校代数的学习和教学．[美]格拉斯主编，陈昌平等译，数学教与学的手册．上海：上海教育出版社，1999：426-427．
[100] Tall, D.O., & Thomas, M. Encouraging versatile thinking in algebra using the computer. Educational Studies in Mathematics, 1991(11): 40-46.
[101] Stacey, K., & MacGregor, M. Algebraic sums and products: Students' concepts and symbolism. In da Ponte, J. P., & Matos, J. F. (eds.), Proceedings of the 18th International Conference for the Psychology of Mathematics Education. Lisbon, Portugal, 1994: 43, 289-296.
[102] Sfard, A. The development of Algebra: Confronting historical and psychological perspectives. Journal of Mathematical Behavior, 1995(14): 15-39.
[103] 张奠宙，李士锜，李俊．数学教育学导论．北京：高等教育出版社，2003：185．
[104] 戴再平．数学习题理论．上海：上海教育出版社，2000：162-185．
[105] 罗增儒．解题分析——谈错例剖析．中学数学教学参考，1999(12)：32-35．
[106] 李善良．数学概念学习中的错误分析．数学教育学报，2002，11(3)：6-11．
[107] 徐文彬．“数学概念学习中的错误分析”之分析．数学教育学报，2004，13(1)：45-47．
[108] 周友士．数学教学中错误概念的诊断与矫治．教育探索，2004(10)：64-66．
[109] 马欣．“潜在假设其及对学生思维的影响”．数学通报，1986(4)：6-9．
[110] Charnay, R. Les enseignements des mathematiques et les erreurs de leurs eleves, Grand N, [Teachers of mathematics and the errors of their pupils]. Institute de Recherche sur 1'Enseignement de Mathematique (IREM) de Grenoble, 1989: 1-21.
[111] Clark, C.M. Asking the right questions about teacher participation: contribution of research on teacher thinking. Educational Researcher, 1988, 17(3), 5-12.
[112] Milhaud, H. Le comportement des maitres face aux erreus des eleves, [Teachers'attitudes towards pupils errors]. DEA [Masters thesis] de Didactique des Mathematiques, IREM de Bordeaux, 1980.
[113] Gagatsis, A., & Christou, C. Errors in mathematics: A multidimensional approach. Scientia Paedagogica Experimentalis, 1997, 34(1): 89-116.
[114] Gagatsis, A., & Kyriakides, L. Teachers' attitudes towards their pupils' mathematical errors. Educational Research and Evaluation, 2000, 6(1): 24-58.
[115] Pundt, J. S. Attitudes toward mathematics, mathematical skills, and error pattern analysis: The relationship among these and other attributes relevant to teaching elementary mathematics, 1990. http://wwwlib.global.umi.com/dissertations.
[116] Mitchell, R.A. The effects of a "buggy" error pattern computer game on preserviee elementary school teachers' diagnostic skills in arithmetic, 1990. http://wwwlib.global.umi.com/dissertations.
[117] Mattair, J. The use of error pattern analysis in the diagnosis and remediation of whole number computational difficulties, 1981. http://wwwlib.global.umi.com/dissertations.
[118] http://www.specialconnections.ku.edu/～specconn/page/instruction/math/pdf/patternanalysis.pdf.
[119] Even, R., & Tirosh, D. Teacher knowledge and understanding of students' mathematical learning. In English, L. (eds.), Handbook of International Research in Mathematics Education. Mahwah, NJ: Erlbaum, 2002:219-240.
[120] Fennema, E., et al. A longitudinal study of learning to use children's thinking in mathematics instruction, Journal for Research in Mathematics Education, 1992(27): 403-434.
[121] Franke, M., Carpenter, T., Levi, L., et al. Capturing Teachers' generative change: a follow-up study of professional development in mathematics, American Educational Research Journal, 2001(38): 653-689.
[122] Fennema, E., & Franke, M.L. Teacher's knowledge and its impact. In Grouws, D.A. (eds), Handbook of Mathematics Teaching and Learning. New York: Macmillan Publishing Company, 1992: 147-164.
[123] Carpenter, T., Fennema, E., & Franke, M. Cognitively guided instruction: a knowledge base for reform in primary mathematics instruction. The Elementary School Journal, 1992(27): 403-434.
[124] Carpenter, T. P., Fennema, E., Peterson, P. L., et al. Using knowledge of children's mathematical thinking in classroom teaching: An experimental study. American Educational Research Journal, 1989(26): 499-532.
[125] Steinberg, R. M., Empson, S. B., & Carpenter, T. P. Inquiry into children's mathematical thinking as a means to teacher change. Journal of Mathematics Teacher Education, 2004(7): 237-267.
[126] Stephens, A. C. Equivalence and relational thinking: preservice elementary teachers' awareness of opportunities and misconceptions. Journal of Mathematics Teacher Education, 2006(9): 249-278.
[127] Tirosh, D. Enhancing prospective teachers' knowledge of children's conceptions: the case of division of fractions. Journal for Research in Mathematics Education, 2000(31): 5-25.
[128] Ball, D.L. Prospective elementary and secondary teachers' understanding of division. Journal for Research in Mathematics Education, 1990, 21(2): 132-144.
[129] Even, R., & Tirosh, D. Subject-matter knowledge and knowledge about students, as sources of teacher presentations of the subject matter. Educational Studies in Mathematics, 1995, 29(1): 1-20.
[130] Tirosh, D., Even, R., & Robinson, N. Simplifying algebraic expressions: teacher awareness and teaching approaches. Educational Studies in Mathematics, 1998(35): 51-64o
[131] Heid, M.K., Blume, G.W., Zbiek, R., et al. Factors that influence teachers learning to do interviews to understand students' mathematical understandings. Educational Studies in Mathematics, 1999(37): 223-249.
[132] Steele, M.D. Comparing knowledge bases and reasoning structures in discussions of mathematics and pedagogy. Journal of Mathematics Teacher Education, 2005(8): 219-328.
[133] Davis, B. Learning for differences: an evolving conception of mathematics teaching. Journal for Research in Mathematics Education, 2000, 28(3): 355-377.
[134] Borko, H., Eisenhart, M., Brown, C.A., et al. Learning to teach hard mathematics: Do novice teachers and their instructors give up too easily? Journal for Research in Mathematics Education, 1992(23):194-222.
[135] Mewborn, D. Reflective thinking among preservice elementary mathematics teachers. Journal for Research in Mathematics Education, 1999(30): 316-341.
[136] Crespo, S. Seeing more than right and wrong answers: prospective teachers' interpretations of students' mathematical work. Journal of Mathematics Teacher Education, 2000(3): 155-181.
[137] Empson, S.B., & Junk, D.L. Teachers' knowledge of children's mathematics after implementing a student-centered curriculum. Journal of Mathematics Teacher Education, 2004(7): 121-144.
[138] Little, J.W. Locating learning in teachers' community of practice: opening up problems of analysis in records of everyday work. Teaching and Teacher Education, 2002(18): 917-946.
[139] Even, R., & Wallach, T. Between student observation and student assessment: a critical reflection. Canadian Journal of Science, Mathematics, and Technology Education, 2004(4): 483-495.
[140] Doerr, H.M. Examining the tasks of teaching when using students' mathematical thinking. Educational Studies in Mathematics, 2006(62): 3-24.
[141] 章建跃．中学生数学学科自我监控能力．上海：华东师范大学出版社，2003：3．
[142] 曾拓，李黎．教师教学能力研究综述．绍兴文理学院学报，2003(1)：102-105．
[143] 申继亮，王凯荣．论教师的教学能力．北京师范大学学报(社科版)，2000(1)：64-71．
[144] [苏]克鲁捷茨斯基，李伯黍等译．中小学生数学能力心理学．上海：上海教育出版社，1983：16，48．
[145] 王宪平．课程改革视野下教师教学能力发展研究．博士学位论文，华东师范大学，2006．
[146] 喻平．数学教育心理学．南宁：广西教育出版社，2004：340-350．
[147] 付敏，刘皴．现代数学教师的能力结构．课程·教材·教法，2005，25(4)：78-82．
[148] 申继亮，辛涛．论教师教学监控能力．北京师范大学学报(社科版)，1995(1)：37-45．
[149] 辛涛，林崇德，申继亮．教师教学监控能力与其教育观念的关系．心理发展与教育，1997，30(2)：36-40．
[150] 申继亮，辛涛．论教师教学监控能力提高的方法和途径．北京师范大学学报(社科版)，1998(1)：35-42．
[151] 辛涛，申继亮，林崇德．教师教学监控能力的结构：一个验证性的研究．心理学报，1998(3)：281-288．
[152] 辛涛，林崇德．教师教学监控能力发展：质与量的分析．中国教育学刊，1999(3)：281-288．
[153] 张景焕，金盛华，陈秀珍．小学教师课堂教学设计能力发展特点及影响因素．心理发展与教育，2004(4)：59-63．
[154] 张景焕，陈秀珍．小学教师关于课堂教学设计能力形成与发展的反思．当代教育科学，2006(17)：30-35．
[155] 曾拓，张彩云，申继亮．中小学教师教学问题诊断能力的研究．教育理论与实践，2005(25)：29-31．
[156] 毛泽东．毛泽东选集(第一卷)．北京：北京人民出版社，1991：286．
[157] [苏]斯托利亚尔，丁尔陞译．数学教育学．北京：人民教育出版社，1984．10．
[158] Borko, H., & Putnam, R.T. Learning to teach. In Berliner, D.C., & Calfee, R.C.(Eds.), Handbook of Educational Psychology. New York: Macmillan, 1996: 673-708.
[159] Clark, C.M., & Yinger, R.J. Teacher planning. In J. Calderhead (ed.), Exploring Teachers' Thinking. London: Cassell, 1987: 84-103.
[160] Connelly, F.M., & Clandinin, D.J. Personal practical knowledge and the modes of knowledge: Relevence for teaching and learning. In E. Eisner(ed.), learning and teaching the ways of knowing (Vol. 84th yearbook of the National Society for the Study of Education, Part Ⅱ., pp.174-198.) Chicago: University of Chicago Press, 1985.
[161] 林崇德．学习与发展．北京：教育出版社，1992：147-149，158．
[162] 查有梁．系统科学与教育．北京：人民教育出版社，1995．
[163] [英]卡尔·波普尔．猜想与反驳．上海：上海译文出版社，1986：199，317．
[164] [美]简·卢文格．自我的发展．杭州：浙江教育出版社，1998：31．
[165] 房剑森．高等教育发展论．桂林：广西师范大学出版社，2001：8-11．
[166] 新华词典编篡组编．新华词典(修订版)．北京：商务印书馆，1989：224．
[167] Rossman, G.B., & Wilson, B.L. Numbers and words: combining quantitative and qualitative methods in a single large-scale evaluation study. Evaluation Review, 1985, 9(5): 627-643.
[168] Cobb, P., Wood, T., Yackel, E., et al. Characteristics of classroom mathematics: an interactional analysis. American Educational Research Journal 1992, 29(3): 573-604.
[169] Pirie, S. Classroom video-recording: When, why and how does it offer a valuable data source for qualitative research? Eric Document Reproduction Service Number ED401128, 1996.
[170] Schoenfeld, A.H. When good teaching leads to bad results: the disasters of "well-taught" mathematics courses. Educational Psychologist, 1988, 23(2): 145-166.
[171] [德]埃德蒙德·胡塞尔．倪梁康译．现象学的方法．上海：上海译文出版社，2005：40．
[172] 马云鹏．课程实施探索．长春：东北师范大学出版社，2001：88．
[173] 郑秋贤．“冲破坚冰”——三位浸入式教师成长的故事．博士学位论文，华东师范大学，2003：12-13．
[174] Stake, E. The Art of Case Study Research. Thousand Oaks: SAGA Publications, 1995.
[175] Zazkis, R. Interviewing in mathematics education research: Choosing the questions. Journal of Mathematical Behavior, 1999, 17(4): 429-439.
[176] 刘曼丽．小数学概念知多少．高雄：高雄复文图书出版社，2002：32-39．
[177] Wilson, P., Cooney, T., & Stinson, D. What constitutes good mathematics teaching and how it develops: nine high school teachers' perspectives. Journal of Mathematics Teacher Education, 2001(8): 83-111.
[178] 陈向明．质的研究方法与社会科学研究．北京：教育科学出版社，2003：388-390．
[179] Stigler, J., & Hiebert, J. Using video surveys to compare classrooms and teaching across cultures: examples and lessons from the TIMSS video studies. Educational Psychologist, 1997, 35(2): 87-100.
[180] 朱德全，宋乃庆．现代教育统计与测评技术．重庆：西南师范大学出版社，1998．
[181] 罗增儒．数学解题学引论．西安：陕西师范大学出版社，2004：499．
[182] 叶澜．教师角色与教师发展新探．北京：教育科学出版社，2001：1．
[183] Sfard, A. On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 1991(22): 1-36.
[184] Sfard, A. On the two metaphors for learning and the dangers of choosing just one. Educational Researcher, 1991, 27(2): 4-13.
[185] 朱德全，宋乃庆，罗万春．数学课程改革与教师教学观念的转变和角色的换．中国教育学刊，2001(12)：37-39．
[186] Shulman, L. Those who understand: Knowledge growth in teaching. Educational Researcher, 1986, 15(2): 4-14.
[187] 张奠宙，赵小平．当心“去数学化”(编后漫笔)．数学教学，2005(6)：封面三．
[188] 陈重穆，宋乃庆．淡化形式，注重实质．数学教育学报，1993(4)：4-9．
[189] Clark, C.M., & Peterson, P.L. Teachers' thought processes. In Wittrock, M.C. (ed.). Handbook of Research on Teaching. Third Edition. New York: Macmillan, 1986.
[190] Ramond, A.M. inconsistency between a beginning elementary school teacher' mathematics beliefs and teaching practice, Journal for Research in Mathematics Education, 1997(28): 550-577.
[191] 程广文．课堂提问研究．博士学位论文，华东师范大学，2002：91．
[192] 中华人民共和国教育部．全日制义务教育数学课程标准(实验稿)．北京：北京师范大学出版社，2001，2．
[193] Mason, J. Enabling teachers to be real teachers: necessary levels of awareness and structure of attention. Journal of Mathematics Teacher Education, 1998(1): 243-267.
[194] 罗增儒．数学解题．重庆：数学教育高级研讨班报告，2006年11月．
[195] Shulman, L. Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 1987, 57(1): 1-22.
[196] Ernest, P. The knowledge, beliefs and attitudes of the mathematics teacher: A model. Journal of Education for Teaching, 1989, 15(1):13-33.
[197] Ma, L. Knowing and Teaching Elementary Mathematics: Teacher's Understanding of Fundamental Mathematics in China and the United States. Mahwah, N.J.: Lawrence Erlbaum Associates, Publishers, 1999.
[198] Vygotsky, L.S. Mind in society. Cambridge,MA: Harvard University Press, 1978.
[199] Arbaugh, F. Study groups as a form of professional development for secondary mathematics teacher. Journal of Mathematics Teacher Education, 2003(6): 139-163.
[200] Cobb, P., & Bowers, J.S. Cognitive and situated learning perspectives in theory and practice. Educational Researcher, 1999, 28(2): 4-15.
[201] 顾泠沅．教育任务与案例分析．上海教育科研，1998(3)：2-6．
[202] Zaslavsky, O. Seizing the opportunity to create uncertainty in learning mathematics. Educational Studies in Mathematics, 2005(60): 297-321.
[203] Zaslavsky, O., Chapman, O., & Leikin, R. Professional development in mathematics education: trends and tasks. Second International Handbook of Mathematics Education. Dordrecht, Boston, London: Kluwer Academic Publishers, 2002: 877-916.
[204] 韩华球．错误：一笔重要的教学资源．课程·教材·教法，2005(3)：26-30．
[2] 戚业国，陈玉琨．学校发展与教师的专业发展．教育理论与实践，2002(8)：27-31．
[3] 联合国教科文组织总部中文科译．教育—财富蕴藏：其中国际21世纪教育委员会报告．北京：教育科学出版社，1996：14-15．
[4] 刘捷．专业化：挑战21世纪的教师．北京：教育科学出版社，2005：66-68．
[5] 范良火．教师知识发展研究．上海：华东师范大学出版社，2003：26-33．
[6] 教育部师范教育司．教师专业化的理论与实践．北京：人民教育出版社，2003：54．
[7] 李其龙，陈永明．教师教育课程的国际比较．北京：教育科学出版社，2002：139-140．
[8] 邵瑞珍．教育心理学—学与教的原理．上海：上海教育出版社，1983：265．
[9] 罗树华，李洪珍．教师能力学．济南：山东教育出版社，1997：27-73．
[10] Brown. H. D. Principles of Language Learning and Teaching. New Jersey: Prentice-Hall, 1987: 204-207.
[11] 魏红，申继亮．中小学教师评价学生能力的现状与对策．教育理论与实践，2004(5)：27-30．
[12] 申继亮，辛涛，邹泓．中小学教师教学能力观的比较研究．教育科学研究，1998(1)：1-4．
[13] 段宝斌，刘猛．对体育教师能力结构的认识．体育研究，2006(7)：163-164．
[14] Brown, J.S., & Burton, R.R. Diagnostic models for procedural bugs in basic mathematical skills. Cognitive Science, 1978(2): 155-192.
[15] 郑毓信，梁贯成．认知科学、建构主义与数学教育．上海：上海教育出版社，1998：62，229．
[16] 全美数学教师理事会著，蔡金法等译．美国数学教育的原则和标准．北京：人民教育出版社，2004：16-17，23-25．
[17] 王梓坤．今日数学及其应用．面向21世纪的数学教育—数学家论数学教育．南京：江苏教育出版社，1994：1-36．
[18] 郑君文，张恩华．数学学习论．南宁：广西教育出版社，2001：93．
[19] 张孝达．学生认为“(－3)×(－4)=9”该怎么办?中小学数学(教师版)，2004(5)：3．
[20] 李道路，孙朝仁．“有理数的乘法”教学实录及评析．中学数学杂志(初中)，2004(6)：15-19．
[21] 许芬英．从有理数乘法的引入看教材的多样化及其对教学的启示．中学数学教育，2004(10)．
[22] 曾飞鹏．关于“(－3)×(－4)=9”．中小学数学(教师版)，2004(7-8)：14-15．
[23] 梁国茂．关于学生认为“(－3)×(－4)=9”的解决办法．中小学数学(教师版)，2004(9)：9-10．
[24] 杨文鹏．读“(－3)×(－4)=9”有感．中小学数学(教师版)，2004(9)：10-11．
[25] 周金华，孙鹏国．关于“(－3)×(－4)=?”的教学．中小学数学(教师版)，2004(10)：6-7．
[26] 张成国．对一个典型案例的再思考．中小学数学(教师版)，2004(11)：7-9．
[27] 郑生．关于“有理数乘法”教学的探讨．中小学数学(教师版)，2004(12)：4-5．
[28] 李学华，申瑞青．读“有理数乘法教学”有感．中小学数学(教师版)，2005(3)：27-28．
[29] 李泽云．也谈“有理数乘法”的教学．中小学数学(教师版)，2005(4)：2-3．
[30] 章如芝．还谈“有理数乘法”的教学．中小学数学(教师版)，2005(7-8)：22-23．
[31] 汲广川．这样的情境不必设置．中小学数学(教师版)，2006(3)：25-26．
[32] 张广祥，张奠宙．代数教学中的模式直观．数学教育学报，2006(1)：1-4．
[33] 罗增儒．案例创作：“(－3)×(－4)=?”数轴表示的挑战．中学数学教学参考，2004(12)：5-7．
[34] 程来平．“1/2＋1/3＝2/5”和“1/2＋1/3＝5/6”具有同等价值．小学教学设计，2005(2)：52．
[35] 孙国元．“1/2＋1/3＝2/5”的价值在哪?小学教学设计，2005(7-8)：107．
[36] 蔡跟东．由“1/2＋1/3＝2/5”和“1/2＋1/3＝5/6”具有同等价值所想到的．小学教学设计，2005(7-8)：107．
[37] 郑毓信．数学教育哲学．成都：四川教育出版社(第二版)，2004：48-68，383，395，325．
[38] 李善良．现代认知观下的数学概念学习与教学．南京：江苏教育出版社，2005：286．
[39] 周友士．数学教学中要善待学生的错误．当代教育科学，2005(3)：52-54．
[40] Hiebert, J., & Carpenter, T.P. Learning and teaching with understanding. In Grouws, D.A. (eds.), Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics. New York: Macmillan, 1992: 65-97.
[41] Newton, D.P. Teaching for understanding. London and New York: Routledge Falmer, 2000.
[42] Ball, D.L. What do students know? Facing challenges of distance, context and desire in trying to hear children. In Biddle, B.J., Good, T.L., & Goodson, I.F. (eds.), International Handbook of Teachers and Teaching. Dordrecht, The Netherlands: Kluwer, 1997: 769-818.
[43] Wallach, T. Hearing students: the complexity of understanding what they are saying, showing, and doing. Journal of Mathematics Teacher Education, 2005(8): 393-417.
[44] Wildy, H., & Wallace, J. Understanding teaching or teaching for understanding. American Educational Research Journal, 1992: 29.
[45] Morgan, C., & Watson, A. The interpretative nature of teachers' assessment of students' mathematics: Issues for equity. Journal for Research in Mathematics Education, 2002(33): 78-110.
[46] Lin, P.J. Conceptualizing teachers' understanding of students' mathematical learning by using assessment tasks. International Journal of Science and Mathematics Education, 2006(4): 545-580.
[47] 朱德全，宋乃庆．论素质教育观下的数学教育．教育研究，1998(5)：23-27．
[48] 罗增儒．“以错纠错”的案例分析．中学数学教学参考，2001(9)：23-26．
[49] ICMI. The fifteenth ICMI study: the professional education and development of teachers of mathematics. Educational Studies in Mathematics, 2004(56): 359-372.
[50] 顾泠沅，易凌峰，聂必凯．寻找中间地带．上海：上海教育出版社，2003：54-164．
[51] Buswell, G.T., & Judd, C.H. Summary of Educational Investigations Relating to Arithmetic (Supplementary Educational Monograph No.27). Chicago: University of Chicago, 1925.
[52] Radatz, R. Error analysis in mathematics education. Journal for Research in Mathematics Education, 1979(5): 163-172.
[53] Mays, M. Misconceptions. In American Perspectives on the Seventh International Congress on Mathematical Education. Reston, Va.: National Council of Teachers of Mathematics, 1993.
[54] Langley, P., Wogulis, J., & Ohlsson, S. Rules and principles in cognitive diagnosis. In Frederiksen, N., Glaser, R., Lesgold, A.M., & Shafto, G. (eds.), Diagnostic Monitoring of Skill and Knowledge Acquisition. Hillsdale, NJ: Lawrence Erlbaum, 1990:217-249.
[55] Tatsuoka, K., & Corter, J. Patterns of diagnosed mathematical content and process skill in TIMSS-R across a sample of 20 countries. American Educational Research Journal, 2004, 41(4): 24-58.
[56] Sleeman, D., Kelly, A.E., Martinak, R., et al. Studies of diagnosis and remediation with high school algebra students. Cognitive Science, 1989(13): 551-568.
[57] Riccomini, P.J. Identification and remediation of systematic error patterns in subtraction. Learning Disability Quaterly, 2005(28): 233-242.
[58] David, S. On mathematical error. Studies in History and Philosophy of Science, 1997, 28(3): 393-415.
[59] Fiori, C., & Zuccheri, L. An experimental research on error patterns in written subtraction. Educational Studies in Mathematics, 2005(60): 323-331.
[60] 彭爱辉．国际数学教育委员会第二届克莱因奖和弗莱登塔尔奖得主一瞥．数学通报2006(11)：56-58．
[61] Bachelard, G. La Formation de l' Esprit Seientifique [The formation of scientific spirit]. Paris: Vrin, 1983.
[62] Brousseau, G. Les obstacles epistemologiques et les problemes en Mathematiques, [Epistemological obstacles and problems in mathematics], Recherches en Didactique des Mathematiques, 1983, 4(2): 164-198.
[63] Greer, B. The mathematics modeling perspective on world problems. Journal of mathematical behaviour, 1993(8): 143-160.
[64] Silver, E.A., Shapiro, L.J., & Deutch, A. Sense making and the solution of division problems involving remainders: an examination of middle school students' solutions processes and their interpretations of solutions. Journal for Research in Mathematics Education, 1993(24): 117-135.
[65] Cai, J., & Silver, E.A. Solution processes and interpretations of solutions in solving a division-with-remainder story problem: Do Chinese and U.S. students have similar difficulties? Journal for Research in Mathematics Education, 1995(26): 491-497.
[66] Brousseau, G. The crucial role of the didactical contract in the analysis and construction of situations in teaching and learning mathematics. In Steiner, H.G. (eds.), Theory of mathematics education. Occasional paper 54. Bielefeld: IDM, 1984: 110-119.
[67] Schoenfeld, A.H. Mathematical Problem Solving. New York: Academic Press, 1985.
[68] D'Amore, B., & Martini, B. Contrat didactiqque, modeles mentaux et modeles intutifs dans la resolution de problemes scolaires standard, [Students' responses to typical school problems with lack of data]. Scientia Paedagogica Experimentalis, 1998, 35(1): 55-94.
[69] Schoenfeld, A.H. What's all the fuss about metacognition? In Schoenfeld, A.H. (eds.), Cognitive Science and Mathematics Education. Hillsdale: Lawrence Erlbaum Ass, 1987: 189-215.
[70] Davis, R. Learning Mathematics: The Cognitive Approach to Mathematics Education. London: Routledge, 1989.
[71] Hald, M. E. A PASS cognitive processes intervention study in mathematics, 2000. http://wwwlib.global.umi.com/dissertations.
[72] Erlwanger, S.H. Benny's conception of rules and answers in IPI mathematics. The Journal of Children's Mathematical Behavior, 1973, 1(2): 7-26.
[73] 李士锜．PME：数学教育心理学．上海：华东师范大学出版社，2001：213-215．
[74] Bock, D.D., Verschaffel, L., & Janssens, D. The predominance of the linear model in secondary school students' solutions of word problems involving length and area of similar plane figures. Educational Studies in Mathematics, 1998(35): 65-83.
[75] Bock, D.D., Dooren, W., Janssens, D., et al. Improper use of linear reasoning: an in-depth study of the nature and the irresistibility of secondary school students' error. Educational Studies in Mathematics, 2002(50): 311-334.
[76] Fischbein, E. Intuition in Science and Mathematics. D.Reidel, Dordrecht. 1987.
[77] Economou, P. How teachers of mathematics confront students' errors. In Philippou, G., Christou, C., & Kakas, A. (eds.), Proceedings of the Second Panhellenic Conference on Mathematics Education and the Informatics in Education. Nicosia: Sighroni Epoxi, 1995: 383-400.
[78] Reason, J. Skill and Error in Everyday Life, Adult Learning: Psychological Research and Applications. New York: Wiley, 1977.
[79] [苏]舍瓦列夫，孙经灏译．学生作业中的概括联想．北京：科学出版社，1963：53．
[80] [苏]孜科娃，孙以芾译．掌握初等几何知识的心理概论．北京：人民教育出版社，1959：37，19-22．
[81] 卢仲衡．初二学生学习数学所产生的一些错误分析．数学通报，1961(7)：29-34．
[82] 卢仲衡，应玉叶，张梅玲等．关于如何减少图形交错的消极影响的问题．数学通报，1964(7)：16-19．
[83] [美]奥苏伯尔(Ausubel)，佘星南，宋钧译．教育心理学——认知观点．北京：人民教育出版社．1994：122-123．
[84] [苏]斯涅普坎，时勘等译．数学教学心理学．重庆：重庆出版社，1987：25．
[85] Tirosh, D., & Stavy, R. Intuitive rules: a way to explain and predict students' reasoning. Educational Studies in Mathematics, 1999(38): 51-69.
[86] Stavy, R. & Tirosh, D. How Students (Mis-)Understand Science and Mathematics: Intuitive Rules. Teachers College Press, 2000.
[87] Stavy, R., Babai, R., Tsamir, P., et al. Are intuitive rules universal? International Journal of Science and Mathematics Education, 2006(4): 417-436.
[88] Dooren, V., Bock, W. D., Weyers, D. D., et al. The predictive power of intuitive rules: A critical analysis of the impact of "more A-more B" and "same A-same B". Educational Studies in Mathematics, 2004(56): 179-207.
[89] Clement, J. Algebra word problem solutions: thought processes underlying a common misconception. Journal for Research in Mathematics Education, 1982, 13(1): 16—30.
[90] Leron, U., & Hazzan, O. The rationality debate: application of cognitive psychology to mathematics education. Educational Studies in Mathematics, 2006(62): 105-126.
[91] Chaiken, S., & Trope, Y. Dual-process Theories in Social Psychology. New York: Guilford, 1999.
[92] Ben, Z.T. The nature and origin of rational errors in arithmetic thinking: Introduction from examples and prior knowledge. Cognitive Science, 1995(19): 341-376.
[93] 吕林海．数学理解学习与教学研究．博士学位论文，华东师范大学，2005：99-109．
[94] Raffiella, B. Capitalizing on errors as "springboards for inquiry": a teaching experiment. Journal for Research in Mathematics Education, 1994, 25(2): 166-208.
[95] O'Connell, A. A. A classification of student errors in probability problem-solving. [Unpublished doctoral dissertation], Teachers College, Columbia University, New York, 1993.
[96] O'Connell, A.A. Understanding the nature of errors in probability problem-solving. Educational Research and Evaluation, 1999, 5(1): 1-21.
[97] 张奠宙主编．数学教育研究导引．南京：江苏教育出版社，1994：331-352．
[98] 张奠宙，宋乃庆．数学教育概论．北京：高等教育出版社，2005：7-8．
[99] Kieran，C．学校代数的学习和教学．[美]格拉斯主编，陈昌平等译，数学教与学的手册．上海：上海教育出版社，1999：426-427．
[100] Tall, D.O., & Thomas, M. Encouraging versatile thinking in algebra using the computer. Educational Studies in Mathematics, 1991(11): 40-46.
[101] Stacey, K., & MacGregor, M. Algebraic sums and products: Students' concepts and symbolism. In da Ponte, J. P., & Matos, J. F. (eds.), Proceedings of the 18th International Conference for the Psychology of Mathematics Education. Lisbon, Portugal, 1994: 43, 289-296.
[102] Sfard, A. The development of Algebra: Confronting historical and psychological perspectives. Journal of Mathematical Behavior, 1995(14): 15-39.
[103] 张奠宙，李士锜，李俊．数学教育学导论．北京：高等教育出版社，2003：185．
[104] 戴再平．数学习题理论．上海：上海教育出版社，2000：162-185．
[105] 罗增儒．解题分析——谈错例剖析．中学数学教学参考，1999(12)：32-35．
[106] 李善良．数学概念学习中的错误分析．数学教育学报，2002，11(3)：6-11．
[107] 徐文彬．“数学概念学习中的错误分析”之分析．数学教育学报，2004，13(1)：45-47．
[108] 周友士．数学教学中错误概念的诊断与矫治．教育探索，2004(10)：64-66．
[109] 马欣．“潜在假设其及对学生思维的影响”．数学通报，1986(4)：6-9．
[110] Charnay, R. Les enseignements des mathematiques et les erreurs de leurs eleves, Grand N, [Teachers of mathematics and the errors of their pupils]. Institute de Recherche sur 1'Enseignement de Mathematique (IREM) de Grenoble, 1989: 1-21.
[111] Clark, C.M. Asking the right questions about teacher participation: contribution of research on teacher thinking. Educational Researcher, 1988, 17(3), 5-12.
[112] Milhaud, H. Le comportement des maitres face aux erreus des eleves, [Teachers'attitudes towards pupils errors]. DEA [Masters thesis] de Didactique des Mathematiques, IREM de Bordeaux, 1980.
[113] Gagatsis, A., & Christou, C. Errors in mathematics: A multidimensional approach. Scientia Paedagogica Experimentalis, 1997, 34(1): 89-116.
[114] Gagatsis, A., & Kyriakides, L. Teachers' attitudes towards their pupils' mathematical errors. Educational Research and Evaluation, 2000, 6(1): 24-58.
[115] Pundt, J. S. Attitudes toward mathematics, mathematical skills, and error pattern analysis: The relationship among these and other attributes relevant to teaching elementary mathematics, 1990. http://wwwlib.global.umi.com/dissertations.
[116] Mitchell, R.A. The effects of a "buggy" error pattern computer game on preserviee elementary school teachers' diagnostic skills in arithmetic, 1990. http://wwwlib.global.umi.com/dissertations.
[117] Mattair, J. The use of error pattern analysis in the diagnosis and remediation of whole number computational difficulties, 1981. http://wwwlib.global.umi.com/dissertations.
[118] http://www.specialconnections.ku.edu/～specconn/page/instruction/math/pdf/patternanalysis.pdf.
[119] Even, R., & Tirosh, D. Teacher knowledge and understanding of students' mathematical learning. In English, L. (eds.), Handbook of International Research in Mathematics Education. Mahwah, NJ: Erlbaum, 2002:219-240.
[120] Fennema, E., et al. A longitudinal study of learning to use children's thinking in mathematics instruction, Journal for Research in Mathematics Education, 1992(27): 403-434.
[121] Franke, M., Carpenter, T., Levi, L., et al. Capturing Teachers' generative change: a follow-up study of professional development in mathematics, American Educational Research Journal, 2001(38): 653-689.
[122] Fennema, E., & Franke, M.L. Teacher's knowledge and its impact. In Grouws, D.A. (eds), Handbook of Mathematics Teaching and Learning. New York: Macmillan Publishing Company, 1992: 147-164.
[123] Carpenter, T., Fennema, E., & Franke, M. Cognitively guided instruction: a knowledge base for reform in primary mathematics instruction. The Elementary School Journal, 1992(27): 403-434.
[124] Carpenter, T. P., Fennema, E., Peterson, P. L., et al. Using knowledge of children's mathematical thinking in classroom teaching: An experimental study. American Educational Research Journal, 1989(26): 499-532.
[125] Steinberg, R. M., Empson, S. B., & Carpenter, T. P. Inquiry into children's mathematical thinking as a means to teacher change. Journal of Mathematics Teacher Education, 2004(7): 237-267.
[126] Stephens, A. C. Equivalence and relational thinking: preservice elementary teachers' awareness of opportunities and misconceptions. Journal of Mathematics Teacher Education, 2006(9): 249-278.
[127] Tirosh, D. Enhancing prospective teachers' knowledge of children's conceptions: the case of division of fractions. Journal for Research in Mathematics Education, 2000(31): 5-25.
[128] Ball, D.L. Prospective elementary and secondary teachers' understanding of division. Journal for Research in Mathematics Education, 1990, 21(2): 132-144.
[129] Even, R., & Tirosh, D. Subject-matter knowledge and knowledge about students, as sources of teacher presentations of the subject matter. Educational Studies in Mathematics, 1995, 29(1): 1-20.
[130] Tirosh, D., Even, R., & Robinson, N. Simplifying algebraic expressions: teacher awareness and teaching approaches. Educational Studies in Mathematics, 1998(35): 51-64o
[131] Heid, M.K., Blume, G.W., Zbiek, R., et al. Factors that influence teachers learning to do interviews to understand students' mathematical understandings. Educational Studies in Mathematics, 1999(37): 223-249.
[132] Steele, M.D. Comparing knowledge bases and reasoning structures in discussions of mathematics and pedagogy. Journal of Mathematics Teacher Education, 2005(8): 219-328.
[133] Davis, B. Learning for differences: an evolving conception of mathematics teaching. Journal for Research in Mathematics Education, 2000, 28(3): 355-377.
[134] Borko, H., Eisenhart, M., Brown, C.A., et al. Learning to teach hard mathematics: Do novice teachers and their instructors give up too easily? Journal for Research in Mathematics Education, 1992(23):194-222.
[135] Mewborn, D. Reflective thinking among preservice elementary mathematics teachers. Journal for Research in Mathematics Education, 1999(30): 316-341.
[136] Crespo, S. Seeing more than right and wrong answers: prospective teachers' interpretations of students' mathematical work. Journal of Mathematics Teacher Education, 2000(3): 155-181.
[137] Empson, S.B., & Junk, D.L. Teachers' knowledge of children's mathematics after implementing a student-centered curriculum. Journal of Mathematics Teacher Education, 2004(7): 121-144.
[138] Little, J.W. Locating learning in teachers' community of practice: opening up problems of analysis in records of everyday work. Teaching and Teacher Education, 2002(18): 917-946.
[139] Even, R., & Wallach, T. Between student observation and student assessment: a critical reflection. Canadian Journal of Science, Mathematics, and Technology Education, 2004(4): 483-495.
[140] Doerr, H.M. Examining the tasks of teaching when using students' mathematical thinking. Educational Studies in Mathematics, 2006(62): 3-24.
[141] 章建跃．中学生数学学科自我监控能力．上海：华东师范大学出版社，2003：3．
[142] 曾拓，李黎．教师教学能力研究综述．绍兴文理学院学报，2003(1)：102-105．
[143] 申继亮，王凯荣．论教师的教学能力．北京师范大学学报(社科版)，2000(1)：64-71．
[144] [苏]克鲁捷茨斯基，李伯黍等译．中小学生数学能力心理学．上海：上海教育出版社，1983：16，48．
[145] 王宪平．课程改革视野下教师教学能力发展研究．博士学位论文，华东师范大学，2006．
[146] 喻平．数学教育心理学．南宁：广西教育出版社，2004：340-350．
[147] 付敏，刘皴．现代数学教师的能力结构．课程·教材·教法，2005，25(4)：78-82．
[148] 申继亮，辛涛．论教师教学监控能力．北京师范大学学报(社科版)，1995(1)：37-45．
[149] 辛涛，林崇德，申继亮．教师教学监控能力与其教育观念的关系．心理发展与教育，1997，30(2)：36-40．
[150] 申继亮，辛涛．论教师教学监控能力提高的方法和途径．北京师范大学学报(社科版)，1998(1)：35-42．
[151] 辛涛，申继亮，林崇德．教师教学监控能力的结构：一个验证性的研究．心理学报，1998(3)：281-288．
[152] 辛涛，林崇德．教师教学监控能力发展：质与量的分析．中国教育学刊，1999(3)：281-288．
[153] 张景焕，金盛华，陈秀珍．小学教师课堂教学设计能力发展特点及影响因素．心理发展与教育，2004(4)：59-63．
[154] 张景焕，陈秀珍．小学教师关于课堂教学设计能力形成与发展的反思．当代教育科学，2006(17)：30-35．
[155] 曾拓，张彩云，申继亮．中小学教师教学问题诊断能力的研究．教育理论与实践，2005(25)：29-31．
[156] 毛泽东．毛泽东选集(第一卷)．北京：北京人民出版社，1991：286．
[157] [苏]斯托利亚尔，丁尔陞译．数学教育学．北京：人民教育出版社，1984．10．
[158] Borko, H., & Putnam, R.T. Learning to teach. In Berliner, D.C., & Calfee, R.C.(Eds.), Handbook of Educational Psychology. New York: Macmillan, 1996: 673-708.
[159] Clark, C.M., & Yinger, R.J. Teacher planning. In J. Calderhead (ed.), Exploring Teachers' Thinking. London: Cassell, 1987: 84-103.
[160] Connelly, F.M., & Clandinin, D.J. Personal practical knowledge and the modes of knowledge: Relevence for teaching and learning. In E. Eisner(ed.), learning and teaching the ways of knowing (Vol. 84th yearbook of the National Society for the Study of Education, Part Ⅱ., pp.174-198.) Chicago: University of Chicago Press, 1985.
[161] 林崇德．学习与发展．北京：教育出版社，1992：147-149，158．
[162] 查有梁．系统科学与教育．北京：人民教育出版社，1995．
[163] [英]卡尔·波普尔．猜想与反驳．上海：上海译文出版社，1986：199，317．
[164] [美]简·卢文格．自我的发展．杭州：浙江教育出版社，1998：31．
[165] 房剑森．高等教育发展论．桂林：广西师范大学出版社，2001：8-11．
[166] 新华词典编篡组编．新华词典(修订版)．北京：商务印书馆，1989：224．
[167] Rossman, G.B., & Wilson, B.L. Numbers and words: combining quantitative and qualitative methods in a single large-scale evaluation study. Evaluation Review, 1985, 9(5): 627-643.
[168] Cobb, P., Wood, T., Yackel, E., et al. Characteristics of classroom mathematics: an interactional analysis. American Educational Research Journal 1992, 29(3): 573-604.
[169] Pirie, S. Classroom video-recording: When, why and how does it offer a valuable data source for qualitative research? Eric Document Reproduction Service Number ED401128, 1996.
[170] Schoenfeld, A.H. When good teaching leads to bad results: the disasters of "well-taught" mathematics courses. Educational Psychologist, 1988, 23(2): 145-166.
[171] [德]埃德蒙德·胡塞尔．倪梁康译．现象学的方法．上海：上海译文出版社，2005：40．
[172] 马云鹏．课程实施探索．长春：东北师范大学出版社，2001：88．
[173] 郑秋贤．“冲破坚冰”——三位浸入式教师成长的故事．博士学位论文，华东师范大学，2003：12-13．
[174] Stake, E. The Art of Case Study Research. Thousand Oaks: SAGA Publications, 1995.
[175] Zazkis, R. Interviewing in mathematics education research: Choosing the questions. Journal of Mathematical Behavior, 1999, 17(4): 429-439.
[176] 刘曼丽．小数学概念知多少．高雄：高雄复文图书出版社，2002：32-39．
[177] Wilson, P., Cooney, T., & Stinson, D. What constitutes good mathematics teaching and how it develops: nine high school teachers' perspectives. Journal of Mathematics Teacher Education, 2001(8): 83-111.
[178] 陈向明．质的研究方法与社会科学研究．北京：教育科学出版社，2003：388-390．
[179] Stigler, J., & Hiebert, J. Using video surveys to compare classrooms and teaching across cultures: examples and lessons from the TIMSS video studies. Educational Psychologist, 1997, 35(2): 87-100.
[180] 朱德全，宋乃庆．现代教育统计与测评技术．重庆：西南师范大学出版社，1998．
[181] 罗增儒．数学解题学引论．西安：陕西师范大学出版社，2004：499．
[182] 叶澜．教师角色与教师发展新探．北京：教育科学出版社，2001：1．
[183] Sfard, A. On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 1991(22): 1-36.
[184] Sfard, A. On the two metaphors for learning and the dangers of choosing just one. Educational Researcher, 1991, 27(2): 4-13.
[185] 朱德全，宋乃庆，罗万春．数学课程改革与教师教学观念的转变和角色的换．中国教育学刊，2001(12)：37-39．
[186] Shulman, L. Those who understand: Knowledge growth in teaching. Educational Researcher, 1986, 15(2): 4-14.
[187] 张奠宙，赵小平．当心“去数学化”(编后漫笔)．数学教学，2005(6)：封面三．
[188] 陈重穆，宋乃庆．淡化形式，注重实质．数学教育学报，1993(4)：4-9．
[189] Clark, C.M., & Peterson, P.L. Teachers' thought processes. In Wittrock, M.C. (ed.). Handbook of Research on Teaching. Third Edition. New York: Macmillan, 1986.
[190] Ramond, A.M. inconsistency between a beginning elementary school teacher' mathematics beliefs and teaching practice, Journal for Research in Mathematics Education, 1997(28): 550-577.
[191] 程广文．课堂提问研究．博士学位论文，华东师范大学，2002：91．
[192] 中华人民共和国教育部．全日制义务教育数学课程标准(实验稿)．北京：北京师范大学出版社，2001，2．
[193] Mason, J. Enabling teachers to be real teachers: necessary levels of awareness and structure of attention. Journal of Mathematics Teacher Education, 1998(1): 243-267.
[194] 罗增儒．数学解题．重庆：数学教育高级研讨班报告，2006年11月．
[195] Shulman, L. Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 1987, 57(1): 1-22.
[196] Ernest, P. The knowledge, beliefs and attitudes of the mathematics teacher: A model. Journal of Education for Teaching, 1989, 15(1):13-33.
[197] Ma, L. Knowing and Teaching Elementary Mathematics: Teacher's Understanding of Fundamental Mathematics in China and the United States. Mahwah, N.J.: Lawrence Erlbaum Associates, Publishers, 1999.
[198] Vygotsky, L.S. Mind in society. Cambridge,MA: Harvard University Press, 1978.
[199] Arbaugh, F. Study groups as a form of professional development for secondary mathematics teacher. Journal of Mathematics Teacher Education, 2003(6): 139-163.
[200] Cobb, P., & Bowers, J.S. Cognitive and situated learning perspectives in theory and practice. Educational Researcher, 1999, 28(2): 4-15.
[201] 顾泠沅．教育任务与案例分析．上海教育科研，1998(3)：2-6．
[202] Zaslavsky, O. Seizing the opportunity to create uncertainty in learning mathematics. Educational Studies in Mathematics, 2005(60): 297-321.
[203] Zaslavsky, O., Chapman, O., & Leikin, R. Professional development in mathematics education: trends and tasks. Second International Handbook of Mathematics Education. Dordrecht, Boston, London: Kluwer Academic Publishers, 2002: 877-916.
[204] 韩华球．错误：一笔重要的教学资源．课程·教材·教法，2005(3)：26-30．