国家课程改革背景下中澳数学教师专业行动能力比较研究
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摘要
从十七大报告到《国家中长期教育改革和发展规划纲要(2010—2020年)》,中国在多个方面体现了国家层面对加强教师队伍建设,提升教师素质的重视。
中国教育部于2011年正式颁布了修订版的《义务教育数学课程标准》。澳大利亚课程、评价与报告管理局(ACARA)于2010年制定了首次在全澳大利业范围推行的国家数学课程。在两国的国家数学课程中,“数与代数”与“统计与概率”都分别作为两个独立的内容领域,且在这两个内容上体现出了相同的理念与要求。在“数与代数”领域,在强调必要的熟练准确的算术计算的同时,更加关注算术学习与代数思维发展的衔接,强调数的学习要为代数的学习作好准备；而在“统计与概率”领域,不再过分强调对理论概率的计算,而是加强对数据的重视,注重发展学生的统计思想与统计素养。
无论在中国还是澳大利亚,在推行国家数学课程改革时,数学教师都是至关重要的因素。因此,本论文旨在研究在中国与澳大利亚这两个不同背景的国家中,数学教师在面对国家课程改革时的应对与表现,探究两国数学教师的相似与不同之处及原因,从而评估教师这一重要因素对国家课程改革的实施的重要影响,并进而反思国家课程改革实施的一些重要问题。本研究的主要内容是在国家课程改革背景下,分析、比较中澳两国数学教师的专业行动能力,即教师基于专业知识与素养作出判断与行动的能力。本研究中是指在面对课程改革时,数学教师结合对特定数学内容(“数与代数”与“统计与概率”,鉴于两国数学课程在这两个领域内要求联结算术与代数学习以及发展统计思想的共同理念,本研究中分别使用“关系性思维”与“统计思想”这两个表述)的认识,深入解读数学课程理念及发展方向,理解学生的数学学习与思考,在此基础上设计具体教学情境与内容的专业能力。主要研究以下几个问题：
(1)通过怎样的理论框架与哪些维度,可以对数学教师专业行动能力进行分析?
(2)中澳两国数学教师在“关系性思维”与“统计思想”这两个内容上的专业行动能力表现如何?有什么相似之处与不同点?
(3)数学教师专业行动能力各维度之间存在着怎样的相互关系?
(4)理论框架中的哪些维度最能体现数学教师的专业行动能力,可以被视为数学教师专业行动能力的关键维度与核心因素?
在“学科教学知识(PCK)”与“数学教学知识(MKT)”的理论框架的基础上,结合教师在面对教育系统改革与学校变革的“专业行动能力(Teacher Capacity)"的研究,考虑中澳两国国家数学课程改革的背景,发展出了本研究的理论框架,从四个维度考察数学教师的专业行动能力：维度A,数学知识；维度B,对官方数学课程理念的理解；维度C,对学生数学学习的理解；维度D,教学设计。针对不同数学内容,每个维度下面各自有四个分析的操作性指标。在研究中,对中澳两国的小学高年级或初中数学教师作了问卷调查,分别在“关系性思维”与“统计思想”两个内容上随机选取120份(中澳各60份)与82份(中澳各41份)有效问卷,中澳样本中初中与小学教师的比例完全对应一致。根据每个维度下各自的四个分析指标,首先对问卷进行了质性分析。在对教师的回答进行双重编码后,利用SPSS19英文版作量化分析。之后在每个内容上各选取10位(中澳各5位)教师进行进一步的访谈并作了深度分析。研究发现了以下结论：
(1)两国数学教帅在“关系性思维”与“统计思想”上的专业行动能力呈现出了非常相似的正态分布,在两个内容上高、中、低的比例大约都是20%、60%、20%。虽然差距并不明显,但两国教师在不同的内容上的表现并不相同,在“关系性思维”上,中国教师在数学知识(维度A)与教学设计(维度D)以及总分上要比澳大利亚教师得分更高；而在“统计思想”上,澳大利亚教师在所有四个维度上都比中国教师稍好。这支持了本论文的研究框架,即需要针对不同的数学内容研究数学教师专业行动能力,不应给出普遍意义上的描述。
(2)在“关系性思维”与“统计思想”这两个内容上,数学教师专业行动能力的四个维度之间的关系略有不同,但总体呈现出了相同的结果。教学设计(维度D)与其它三个维度之中的任意一个都表现出了非常显著的统计意义上的相关性。若不考虑教学设计这一维度,其余三个维度之间的相关性并不显著。
(3)本研究理论框架中的四个维度都能有效区分数学教师的专业行动能力,但其中教学设计维度表现最为直接明显。首先,教学设计这一维度对于不同层次水平专业行动能力教师的区分最为显著,线性多远逐步回归分析(step-wize)显示这一维度为回归模型中的首要维度；其次,教学设计这一维度与其它维度的相关性最为密切,而且其它三个维度之间的相关也都是通过教学设计这一维度的作用才发生；最后,两国教师在教学设计上的表现最不理想,要提升他们的专业行动能力,教学设计是需要得到最大重视的一个维度。因此,可以认为教学设计是教师专业行动能力的关键维度与核心要素。
本论文共包含六章：第1章为引论,介绍了研究背景、研究问题、研究目的以及研究意义；第2章为文献综述,在界定了本研究中“教师专业行动能力”的内涵后,对国内外已有的相关研究做了述评。在文献述评中首先回顾了与本研究关系密切的数学教师知识研究的发展及各种研究路径,指出了已有研究的局限性；接着探讨了在本研究中如何从上述研究中发展出“数学教师专业行动能力”的理论框架；第3章为研究设计与方法,首先给出了研究理论框架,之后讨论了研究的设计及实施过程；第4章与第5章是本研究的主要研究内容与发现,分别关注“关系性思维”与“统计思想”这两部分内容,这两章采用完全相同的结构：首先讨论了课程与教学新的理念与要求,接着详细介绍研究过程,之后是对研究问卷进行质性和量化的分析并对研究效度与信度进行了严格的验证,再对部分教师的访谈进行深度分析,最后对研究发现加以总结；第6章是本论文的研究结论与思考,对前面两章的研究发现进行总结与反思,讨论了研究存在的不足与后续研究的设想。
本研究的拟创新之处首先在于首次在数学教育研究中使用“数学教师专业行动能力”这一表述并进行了理论框架的架构,框架中使用了四个维度及详细分析指标。与之前的数学教师知识的研究不同,本研究认为对数学教师的研究应该关注特定的数学内容,并且要关注教师针对特定数学内容的具体教学设计。而且,考虑中澳两国国家数学课程这一背景,将对官方数学课程理念的解读作为理论框架中的一个重要维度。并用实证的方法对这一框架进行了验证与探索。
其次,本研究的创新之处还在于首次系统地对中澳两国数学教师进行了比较。中澳两国都颁布了最新的国家数学课程,而两国的社会文化背景与教育系统差异明显,因此这一比较研究可以为深入了解教师对数学课程改革的推行与实施的重要作用提供启示。
本研究的不足之处主要体现在以下方面：
针对特定内容描述数学教师专业行动能力,这是本研究在理论发展上的尝试与创新,但也存在一定的局限性。“数学教学知识”的理论框架更易于描述一个教师的整体知识状况,而本研究框架在如何更有效地描述某一位教师的整体专业行动能力将存在困难。研究中选取了5位教师同时参与两个内容的研究,发现了同一教师在不同数学内容上表现的一些关联,但结论有限。
另一不足之处体现在研究结论的分析上,本研究同时采用了质性与量化分析,质性与量化分析的结果存在高度的一致性,并通过一系列严格的测试检验了问卷分析的效度与信度。但是对数学教师专业行动能力三个层次的划分难以完全避免主观成分。
此外,本研究没有观察实际课堂教学。虽然本研究中问卷与访谈很好地反映了样本教师的专业行动能力,但没能进一步研究教师的教学设计与实际教学之问的关联。
From the plenary report of17th CPC National Congress to "Medium and Longterm Planning Outline for China's Educational Reform and Development (2010-2020)", China shows its value on improving teachers'quality and enhancing construction of the ranks of teachers at the national level in various aspects.
Ministry of Education of China published the revised "Mathematics Curriculum Standard for Compulsory Education" in2011. Australian Curriculum, Assessment, and Reporting Authority (ACARA) shaped the Australian Curriculum in Mathematics being used across all states and territories in Australia for the first time in2010. In both Chinese and Australian national mathematics curricula,"number and algebra"'and "statistics and probability" are two single out of four (three) content strands. The two curricula in both countries have the same intentions and requirements on these two contents. For "number and algebra", it is more focused on bridging the learning of arithmetic and development of early algebraic thinking, and it is emphasized that the learning of number should be prepared for the leaning of algebra in later years whilst value on necessarily proficient and quick arithmetic calculation still remains. In the content of "statistics and probability", in contrast to traditional approaches to teaching focus on computations of theoretical probability, new emphases on data and development of statistical thinking and literacy are characterized.
The intentions and requirements are clear on the two content strands in both countries. However mathematics teacher is the key element in the curriculum reform and implementation in both China and Australia. To introduce the national mathematics curriculum, several important questions need to be considered such as "whether the teachers have corresponding capacity" and "how to improve teachers'relevant capacity".
As a result, this thesis aimed to look at mathematics teachers' responses and performance when facing the national curriculum reform in China and Australia which have completely different contexts, and to explore the similarities and differences between Chinese and Australian mathematics teachers and the causes. And then to evaluate the important influence to national curriculum reform from teacher as a key element and to think of some big issues on national curriculum reform and implementation. Based on these considerations, the main focus of this research is on teacher capacity of Chinese and Australian teachers under the background of national curriculum reform. That is teachers' judgement and disposition to act based on professional knowledge and experience. In this thesis, it means mathematics teachers' professional ability, when facing the national curriculum reform, to focus on specific mathematical contents ("number and algebra" and "statistics and probability", considering the same intentions on these two contents in both countries-connecting the learning of arithmetic and algebra and developing statistical thinking-,"relational thinking" and "statistical thinking" were respectively used in this thesis), to deeply interpret the intentions and big ideas of mathematics curriculum and to understanding students' mathematical learning and thinking, then to design specific teaching scenario and examples in mathematics classroom. The following questions were intended to be explored:
1. What theoretical framework with what criteria can be used to both qualitatively and quantitatively analyze and classify the capacity between Chinese and Australian teachers?
2. How is the capacity for Chinese and Australian teachers on the two contents of "relational thinking" and "statistical thinking"? What are the similarities and differences?
3. What are the relationships between the different criteria of teacher capacity in mathematics?
4. What criterion (criteria) can best describe teacher capacity in mathematics and can be seen as the critical criterion (criteria) and key element(s)?
Pedagogical Content Knowledge (PCK) initialed by Shulman and Mathematical Knowledge for Teaching (MKT) developed by Ball et al. underpinned the theoretical position of this research. On the basis of PCK and MKT, combined with the research of Teacher Capacity that how teachers face and respond educational system reform and school change, considering contexts of national curriculum reform in both China and Australia, theoretical framework of this research was developed by considering teacher capacity in mathematics from four criteria:Criterion A, Knowledge of Mathematics; Criterion B, Interpretation of Intentions of the Official Mathematics Curriculum; Criterion C, Understanding of Students'Learning and Criterion D, Design of Teaching. For research, selected Chinese and Australian mathematics teachers in upper primary or secondary years were given research questionnaires.120(60Chinese and60Australian) and82(41Chinese and41Australian) were respectively collected for "relational thinking" and "statistical thinking" from the valid research questionnaires, the ratio between secondary teachers and that of primary schools were completely the same in China and Australia sample. Based on the four indicators in each of the four criteria, the collected questionnaires were firstly analyzed qualitatively. After that, all the questionnaires were double-coded according to teachers'responses and quantitative analysis was conducted with SPSS19English version. After the analyses on questionnaires,10teachers (five on "relational thinking" and five on "statistical thinking") from each country-20in total-were selected for the later stage in-depth interview considering their responses to the questionnaire. Several conclusions were found in as following:
1. Teacher Capacity of Chinese and Australian teachers show evidently similar normal distributions on both "relational thinking" and "statistical thinking". The proportion between the numbers of High, Medium and Low Capacity teachers was approximately20%,60%and20%in both contents. However, Chinese and Australian teachers performed differently in the two contents, in "relational thinking", Chinese teachers scored higher on Criterion A (Knowledge of Mathematics) and Criterion D (Design of Teaching), and the global score as well; but in "statistical thinking", Australian teachers performed slightly better than their Chinese counterparts on all four criteria. However, the difference between China and Australia sample was not significant. This point supported the theoretical framework of this thesis which argued that Teacher Capacity should be demonstrated focusing on specific mathematical content, but not just a general description.
2. The interrelations between the four criteria of Teacher Capacity had minor difference, but still showed the same trend. That is Criterion D (Design of Teaching) had statistically significant correlations with any of the other three criteria. However, when separated from Criterion D, the inter-relationships between the other three criteria will not be as strong as their relationship to Criterion D (design of Teaching).
3. All four criteria were effective in classifying teachers, but Criterion D (Design of Teaching) was by far the most powerful criterion. First, Design of Teaching performed most effectively in discriminating between teachers as High, Medium and Low Capacity. Second, among all the four criteria, Design of Teaching has the closest relationships with the other three criteria, and the inter-relationships between the other three criteria are under the associations with Design of Teaching. Finally, in both "relational thinking" and "statistical thinking", teachers in both countries had biggest difficulties and performed most weakly on Design of Teaching, and this is the criterion the teachers need to pay most attention for improving their Teacher Capacity. As a result, Design of Teaching can be considered as the critical dimension and key element of Teacher Capacity.
This thesis comprises of six chapters. Chapter One is Introduction which includes background of research, research questions, aims of research and significance of research. The second chapter is Literature Review, after defining the connotation of Teacher Capacity in this research, this chapter reviews the relevant research in China and in the west. In the review of the literatures, firstly it looks back to the developmental history of relevant research and different research thrusts on "mathematical knowledge for teaching" which has close connections with this research and point out the limitations of existed research. Then it explores how to develop the theoretical framework of Teacher Capacity in mathematics from previous research in this research. Chapter Three is Design and Methodology of Research, theoretical framework is firstly given. Then discuss the research design and process needed to put in effect and the issues of validity and reliability. Main research content and findings of this research are in Chapter Four and Chapter Five which apply the same structure-the new intentions and requirements of instruction for specific mathematical content (relational thinking or statistical thinking) are discussed in the beginning, after that, it introduces the research methodology and process in details, then the qualitative and quantitative analyses of the questionnaires followed by the in-depth interview with selected teachers, finally is the conclusion part of the research findings. The sixth chapter is Research Conclusions and Discussions, it is the summing up and reflections to the previous two chapters and some discussions of the limitations of this research and suggestions of ongoing research.
What is innovative in this research is at first at the construction of the theoretical framework of Teacher Capacity in mathematics. Being different from existing research on mathematics teachers' knowledge, it is argued in this research that the research to mathematics teachers should be focused on specific mathematical contents and look at specific teaching design on particular contents. What is more, considering the background of national curriculum reform in both China and Australia, it takes Interpretation of Intentions of the Official Mathematics Curriculum as one important criterion.
The second innovative point is that this is the first systemic comparative study between Chinese and Australian teachers in mathematics. Both China and Australia have recently published the new national curriculum in mathematics that is mandatory, but there is big difference between the social and cultural contexts and educational system in these two countries. Thus, this comparative study is able to offer valuable implications for deeply understanding the introduction and implementation of curriculum reform in mathematics.
However, certain limitations still remain in this research.
First of all, it is the attempt and innovation on theoretical development of this research to describe Teacher Capacity in mathematics focusing on specific mathematical contents, but meanwhile there exists limitations in this theoretical model. To have comparisons with the research on "Mathematical Knowledge for Teaching (MKT)", it is easier to give an overall classifications based on the teacher's MKT, and there will be difficulties in effectively describe a teacher's global Teacher Capacity using the theoretical model in this research. There were five teachers who were asked to anticipate both research studies on 'relational thinking" and "statistical thinking", some connections were found on one teacher's performance on different mathematical contents, but the conclusions were just implied and limited.
Another limitation of this research was shown in the analysis of the research conclusions. Both qualitative and quantitative research were used in the thesis and the consistency of the qualitative and quantitative research was highly reached. And a serious of rigorous tests were conducted to check the validity and reliability of the analyses of questionnaire. However, the classification of high, medium and low levels for Teacher Capacity in mathematics were arbitrary to some extent.
Furthermore, there was no classroom observation and recordings used in this research. Although the research questionnaires and interviews were robust enough to reflect anticipating teachers' capacity, no further explorations were conducted on the connection between teachers' teaching design and practical instructions.
中国教育部于2011年正式颁布了修订版的《义务教育数学课程标准》。澳大利亚课程、评价与报告管理局(ACARA)于2010年制定了首次在全澳大利业范围推行的国家数学课程。在两国的国家数学课程中,“数与代数”与“统计与概率”都分别作为两个独立的内容领域,且在这两个内容上体现出了相同的理念与要求。在“数与代数”领域,在强调必要的熟练准确的算术计算的同时,更加关注算术学习与代数思维发展的衔接,强调数的学习要为代数的学习作好准备；而在“统计与概率”领域,不再过分强调对理论概率的计算,而是加强对数据的重视,注重发展学生的统计思想与统计素养。
无论在中国还是澳大利亚,在推行国家数学课程改革时,数学教师都是至关重要的因素。因此,本论文旨在研究在中国与澳大利亚这两个不同背景的国家中,数学教师在面对国家课程改革时的应对与表现,探究两国数学教师的相似与不同之处及原因,从而评估教师这一重要因素对国家课程改革的实施的重要影响,并进而反思国家课程改革实施的一些重要问题。本研究的主要内容是在国家课程改革背景下,分析、比较中澳两国数学教师的专业行动能力,即教师基于专业知识与素养作出判断与行动的能力。本研究中是指在面对课程改革时,数学教师结合对特定数学内容(“数与代数”与“统计与概率”,鉴于两国数学课程在这两个领域内要求联结算术与代数学习以及发展统计思想的共同理念,本研究中分别使用“关系性思维”与“统计思想”这两个表述)的认识,深入解读数学课程理念及发展方向,理解学生的数学学习与思考,在此基础上设计具体教学情境与内容的专业能力。主要研究以下几个问题：
(1)通过怎样的理论框架与哪些维度,可以对数学教师专业行动能力进行分析?
(2)中澳两国数学教师在“关系性思维”与“统计思想”这两个内容上的专业行动能力表现如何?有什么相似之处与不同点?
(3)数学教师专业行动能力各维度之间存在着怎样的相互关系?
(4)理论框架中的哪些维度最能体现数学教师的专业行动能力,可以被视为数学教师专业行动能力的关键维度与核心因素?
在“学科教学知识(PCK)”与“数学教学知识(MKT)”的理论框架的基础上,结合教师在面对教育系统改革与学校变革的“专业行动能力(Teacher Capacity)"的研究,考虑中澳两国国家数学课程改革的背景,发展出了本研究的理论框架,从四个维度考察数学教师的专业行动能力：维度A,数学知识；维度B,对官方数学课程理念的理解；维度C,对学生数学学习的理解；维度D,教学设计。针对不同数学内容,每个维度下面各自有四个分析的操作性指标。在研究中,对中澳两国的小学高年级或初中数学教师作了问卷调查,分别在“关系性思维”与“统计思想”两个内容上随机选取120份(中澳各60份)与82份(中澳各41份)有效问卷,中澳样本中初中与小学教师的比例完全对应一致。根据每个维度下各自的四个分析指标,首先对问卷进行了质性分析。在对教师的回答进行双重编码后,利用SPSS19英文版作量化分析。之后在每个内容上各选取10位(中澳各5位)教师进行进一步的访谈并作了深度分析。研究发现了以下结论：
(1)两国数学教帅在“关系性思维”与“统计思想”上的专业行动能力呈现出了非常相似的正态分布,在两个内容上高、中、低的比例大约都是20%、60%、20%。虽然差距并不明显,但两国教师在不同的内容上的表现并不相同,在“关系性思维”上,中国教师在数学知识(维度A)与教学设计(维度D)以及总分上要比澳大利亚教师得分更高；而在“统计思想”上,澳大利亚教师在所有四个维度上都比中国教师稍好。这支持了本论文的研究框架,即需要针对不同的数学内容研究数学教师专业行动能力,不应给出普遍意义上的描述。
(2)在“关系性思维”与“统计思想”这两个内容上,数学教师专业行动能力的四个维度之间的关系略有不同,但总体呈现出了相同的结果。教学设计(维度D)与其它三个维度之中的任意一个都表现出了非常显著的统计意义上的相关性。若不考虑教学设计这一维度,其余三个维度之间的相关性并不显著。
(3)本研究理论框架中的四个维度都能有效区分数学教师的专业行动能力,但其中教学设计维度表现最为直接明显。首先,教学设计这一维度对于不同层次水平专业行动能力教师的区分最为显著,线性多远逐步回归分析(step-wize)显示这一维度为回归模型中的首要维度；其次,教学设计这一维度与其它维度的相关性最为密切,而且其它三个维度之间的相关也都是通过教学设计这一维度的作用才发生；最后,两国教师在教学设计上的表现最不理想,要提升他们的专业行动能力,教学设计是需要得到最大重视的一个维度。因此,可以认为教学设计是教师专业行动能力的关键维度与核心要素。
本论文共包含六章：第1章为引论,介绍了研究背景、研究问题、研究目的以及研究意义；第2章为文献综述,在界定了本研究中“教师专业行动能力”的内涵后,对国内外已有的相关研究做了述评。在文献述评中首先回顾了与本研究关系密切的数学教师知识研究的发展及各种研究路径,指出了已有研究的局限性；接着探讨了在本研究中如何从上述研究中发展出“数学教师专业行动能力”的理论框架；第3章为研究设计与方法,首先给出了研究理论框架,之后讨论了研究的设计及实施过程；第4章与第5章是本研究的主要研究内容与发现,分别关注“关系性思维”与“统计思想”这两部分内容,这两章采用完全相同的结构：首先讨论了课程与教学新的理念与要求,接着详细介绍研究过程,之后是对研究问卷进行质性和量化的分析并对研究效度与信度进行了严格的验证,再对部分教师的访谈进行深度分析,最后对研究发现加以总结；第6章是本论文的研究结论与思考,对前面两章的研究发现进行总结与反思,讨论了研究存在的不足与后续研究的设想。
本研究的拟创新之处首先在于首次在数学教育研究中使用“数学教师专业行动能力”这一表述并进行了理论框架的架构,框架中使用了四个维度及详细分析指标。与之前的数学教师知识的研究不同,本研究认为对数学教师的研究应该关注特定的数学内容,并且要关注教师针对特定数学内容的具体教学设计。而且,考虑中澳两国国家数学课程这一背景,将对官方数学课程理念的解读作为理论框架中的一个重要维度。并用实证的方法对这一框架进行了验证与探索。
其次,本研究的创新之处还在于首次系统地对中澳两国数学教师进行了比较。中澳两国都颁布了最新的国家数学课程,而两国的社会文化背景与教育系统差异明显,因此这一比较研究可以为深入了解教师对数学课程改革的推行与实施的重要作用提供启示。
本研究的不足之处主要体现在以下方面：
针对特定内容描述数学教师专业行动能力,这是本研究在理论发展上的尝试与创新,但也存在一定的局限性。“数学教学知识”的理论框架更易于描述一个教师的整体知识状况,而本研究框架在如何更有效地描述某一位教师的整体专业行动能力将存在困难。研究中选取了5位教师同时参与两个内容的研究,发现了同一教师在不同数学内容上表现的一些关联,但结论有限。
另一不足之处体现在研究结论的分析上,本研究同时采用了质性与量化分析,质性与量化分析的结果存在高度的一致性,并通过一系列严格的测试检验了问卷分析的效度与信度。但是对数学教师专业行动能力三个层次的划分难以完全避免主观成分。
此外,本研究没有观察实际课堂教学。虽然本研究中问卷与访谈很好地反映了样本教师的专业行动能力,但没能进一步研究教师的教学设计与实际教学之问的关联。
From the plenary report of17th CPC National Congress to "Medium and Longterm Planning Outline for China's Educational Reform and Development (2010-2020)", China shows its value on improving teachers'quality and enhancing construction of the ranks of teachers at the national level in various aspects.
Ministry of Education of China published the revised "Mathematics Curriculum Standard for Compulsory Education" in2011. Australian Curriculum, Assessment, and Reporting Authority (ACARA) shaped the Australian Curriculum in Mathematics being used across all states and territories in Australia for the first time in2010. In both Chinese and Australian national mathematics curricula,"number and algebra"'and "statistics and probability" are two single out of four (three) content strands. The two curricula in both countries have the same intentions and requirements on these two contents. For "number and algebra", it is more focused on bridging the learning of arithmetic and development of early algebraic thinking, and it is emphasized that the learning of number should be prepared for the leaning of algebra in later years whilst value on necessarily proficient and quick arithmetic calculation still remains. In the content of "statistics and probability", in contrast to traditional approaches to teaching focus on computations of theoretical probability, new emphases on data and development of statistical thinking and literacy are characterized.
The intentions and requirements are clear on the two content strands in both countries. However mathematics teacher is the key element in the curriculum reform and implementation in both China and Australia. To introduce the national mathematics curriculum, several important questions need to be considered such as "whether the teachers have corresponding capacity" and "how to improve teachers'relevant capacity".
As a result, this thesis aimed to look at mathematics teachers' responses and performance when facing the national curriculum reform in China and Australia which have completely different contexts, and to explore the similarities and differences between Chinese and Australian mathematics teachers and the causes. And then to evaluate the important influence to national curriculum reform from teacher as a key element and to think of some big issues on national curriculum reform and implementation. Based on these considerations, the main focus of this research is on teacher capacity of Chinese and Australian teachers under the background of national curriculum reform. That is teachers' judgement and disposition to act based on professional knowledge and experience. In this thesis, it means mathematics teachers' professional ability, when facing the national curriculum reform, to focus on specific mathematical contents ("number and algebra" and "statistics and probability", considering the same intentions on these two contents in both countries-connecting the learning of arithmetic and algebra and developing statistical thinking-,"relational thinking" and "statistical thinking" were respectively used in this thesis), to deeply interpret the intentions and big ideas of mathematics curriculum and to understanding students' mathematical learning and thinking, then to design specific teaching scenario and examples in mathematics classroom. The following questions were intended to be explored:
1. What theoretical framework with what criteria can be used to both qualitatively and quantitatively analyze and classify the capacity between Chinese and Australian teachers?
2. How is the capacity for Chinese and Australian teachers on the two contents of "relational thinking" and "statistical thinking"? What are the similarities and differences?
3. What are the relationships between the different criteria of teacher capacity in mathematics?
4. What criterion (criteria) can best describe teacher capacity in mathematics and can be seen as the critical criterion (criteria) and key element(s)?
Pedagogical Content Knowledge (PCK) initialed by Shulman and Mathematical Knowledge for Teaching (MKT) developed by Ball et al. underpinned the theoretical position of this research. On the basis of PCK and MKT, combined with the research of Teacher Capacity that how teachers face and respond educational system reform and school change, considering contexts of national curriculum reform in both China and Australia, theoretical framework of this research was developed by considering teacher capacity in mathematics from four criteria:Criterion A, Knowledge of Mathematics; Criterion B, Interpretation of Intentions of the Official Mathematics Curriculum; Criterion C, Understanding of Students'Learning and Criterion D, Design of Teaching. For research, selected Chinese and Australian mathematics teachers in upper primary or secondary years were given research questionnaires.120(60Chinese and60Australian) and82(41Chinese and41Australian) were respectively collected for "relational thinking" and "statistical thinking" from the valid research questionnaires, the ratio between secondary teachers and that of primary schools were completely the same in China and Australia sample. Based on the four indicators in each of the four criteria, the collected questionnaires were firstly analyzed qualitatively. After that, all the questionnaires were double-coded according to teachers'responses and quantitative analysis was conducted with SPSS19English version. After the analyses on questionnaires,10teachers (five on "relational thinking" and five on "statistical thinking") from each country-20in total-were selected for the later stage in-depth interview considering their responses to the questionnaire. Several conclusions were found in as following:
1. Teacher Capacity of Chinese and Australian teachers show evidently similar normal distributions on both "relational thinking" and "statistical thinking". The proportion between the numbers of High, Medium and Low Capacity teachers was approximately20%,60%and20%in both contents. However, Chinese and Australian teachers performed differently in the two contents, in "relational thinking", Chinese teachers scored higher on Criterion A (Knowledge of Mathematics) and Criterion D (Design of Teaching), and the global score as well; but in "statistical thinking", Australian teachers performed slightly better than their Chinese counterparts on all four criteria. However, the difference between China and Australia sample was not significant. This point supported the theoretical framework of this thesis which argued that Teacher Capacity should be demonstrated focusing on specific mathematical content, but not just a general description.
2. The interrelations between the four criteria of Teacher Capacity had minor difference, but still showed the same trend. That is Criterion D (Design of Teaching) had statistically significant correlations with any of the other three criteria. However, when separated from Criterion D, the inter-relationships between the other three criteria will not be as strong as their relationship to Criterion D (design of Teaching).
3. All four criteria were effective in classifying teachers, but Criterion D (Design of Teaching) was by far the most powerful criterion. First, Design of Teaching performed most effectively in discriminating between teachers as High, Medium and Low Capacity. Second, among all the four criteria, Design of Teaching has the closest relationships with the other three criteria, and the inter-relationships between the other three criteria are under the associations with Design of Teaching. Finally, in both "relational thinking" and "statistical thinking", teachers in both countries had biggest difficulties and performed most weakly on Design of Teaching, and this is the criterion the teachers need to pay most attention for improving their Teacher Capacity. As a result, Design of Teaching can be considered as the critical dimension and key element of Teacher Capacity.
This thesis comprises of six chapters. Chapter One is Introduction which includes background of research, research questions, aims of research and significance of research. The second chapter is Literature Review, after defining the connotation of Teacher Capacity in this research, this chapter reviews the relevant research in China and in the west. In the review of the literatures, firstly it looks back to the developmental history of relevant research and different research thrusts on "mathematical knowledge for teaching" which has close connections with this research and point out the limitations of existed research. Then it explores how to develop the theoretical framework of Teacher Capacity in mathematics from previous research in this research. Chapter Three is Design and Methodology of Research, theoretical framework is firstly given. Then discuss the research design and process needed to put in effect and the issues of validity and reliability. Main research content and findings of this research are in Chapter Four and Chapter Five which apply the same structure-the new intentions and requirements of instruction for specific mathematical content (relational thinking or statistical thinking) are discussed in the beginning, after that, it introduces the research methodology and process in details, then the qualitative and quantitative analyses of the questionnaires followed by the in-depth interview with selected teachers, finally is the conclusion part of the research findings. The sixth chapter is Research Conclusions and Discussions, it is the summing up and reflections to the previous two chapters and some discussions of the limitations of this research and suggestions of ongoing research.
What is innovative in this research is at first at the construction of the theoretical framework of Teacher Capacity in mathematics. Being different from existing research on mathematics teachers' knowledge, it is argued in this research that the research to mathematics teachers should be focused on specific mathematical contents and look at specific teaching design on particular contents. What is more, considering the background of national curriculum reform in both China and Australia, it takes Interpretation of Intentions of the Official Mathematics Curriculum as one important criterion.
The second innovative point is that this is the first systemic comparative study between Chinese and Australian teachers in mathematics. Both China and Australia have recently published the new national curriculum in mathematics that is mandatory, but there is big difference between the social and cultural contexts and educational system in these two countries. Thus, this comparative study is able to offer valuable implications for deeply understanding the introduction and implementation of curriculum reform in mathematics.
However, certain limitations still remain in this research.
First of all, it is the attempt and innovation on theoretical development of this research to describe Teacher Capacity in mathematics focusing on specific mathematical contents, but meanwhile there exists limitations in this theoretical model. To have comparisons with the research on "Mathematical Knowledge for Teaching (MKT)", it is easier to give an overall classifications based on the teacher's MKT, and there will be difficulties in effectively describe a teacher's global Teacher Capacity using the theoretical model in this research. There were five teachers who were asked to anticipate both research studies on 'relational thinking" and "statistical thinking", some connections were found on one teacher's performance on different mathematical contents, but the conclusions were just implied and limited.
Another limitation of this research was shown in the analysis of the research conclusions. Both qualitative and quantitative research were used in the thesis and the consistency of the qualitative and quantitative research was highly reached. And a serious of rigorous tests were conducted to check the validity and reliability of the analyses of questionnaire. However, the classification of high, medium and low levels for Teacher Capacity in mathematics were arbitrary to some extent.
Furthermore, there was no classroom observation and recordings used in this research. Although the research questionnaires and interviews were robust enough to reflect anticipating teachers' capacity, no further explorations were conducted on the connection between teachers' teaching design and practical instructions.
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