本研究以合作成長小組的互動做為促進參與教師數學教學專業成長的策略。本研究的目的在針對參與教師個人與合作成長小組的互動改變狀況，探討參與教師數學教學知能及反思能力之成長歷程及其成因。最後建立數學教師的學習模型。
本研究採詮釋性研究法。研究者和台北市一所公立小學的三位一年級教師共同組成數學教學合作成長小組，進行一學年合作的介入研究。三位教師有二個特色，其一是具有成長動機的同校同學年的教師群，另一則是教學信念、教學經驗及人格特質不相同的教師。在研究的過程中，研究者扮演參與觀察者、促進者及探究者的角色。合作成長小組利用每週半天的時間進行教學研討。活動的內容隨著參與教師的學習需求、意願、不足之處及共同形成的目標而做調整。活動內容包括教學活動設計、學生解題類型分析、數學教學案例的討論、教學問題的討論、文獻的研討、教學影帶評析等，最後共同完成一份行動研究報告。
本研究蒐集的資料包括教師教學錄影與錄音、合作成長小組教學研討錄音、教師設計的學習單、反思札記、e-mail、教學研討文件、對學生的解題類型分析、教學計畫、行動研究報告、學生的學習單、學生的施測結果、研究者的札記等。訪談對象包括參與教師、兩位實習教師、學生、行政人員等。所有錄音和錄影資料皆轉錄成逐字稿。本研究遵循Cobb and Whitenack(1996)分析大量質性資料之方法學分析資料。就合作成長小組的互動分析而言，本研究由量的分析瞭解教師互動的概況，由質的分析深入探討教師互動的內涵。研究者選出明顯反應心理和社會互動關係的非等價事件，據此統計合作成長小組互動的交互影響次數及百分比，並將之視覺化。就數學教學知能與反思能力成長的分析而言，本研究分別依據Franke, Carpenter, Levi, and Fennema (2001)的教師發展層次架構及Ward and McCotter (2004)的反思等級架構分析。本研究採多重資料來源、多重資料蒐集程序、多重時間及不同分析者的三角校正，以確保資料的信效度。
本研究由實徵資料歸納出區分教師學習類型的五個範疇：知識類型、學習教學的思維模式、教學知與行的關係、教學取向、數學教學信念。此五範疇及教學知能與反思層次架構將數學教師區分成四種學習類型：素樸型、理論型、實務型、實踐型。成為實踐型教師是教師專業發展的目標。
本研究以活動理論和實作認知理論為理論基礎，由實徵資料形成了兩個模型。其一為數學教師個人與合作成長小組的互動成長模型。製造認知衝突、鼓勵實驗驗證是促進教師專業成長的重要機制。合作成長小組的外在刺激促進教師個人進行社會性反思及教學實驗。教師個人對教學實作做自我性或社會性反思，產生新的知識或信念；再根據該知識或信念透過自我性或社會性反思，實踐於教學中。然後再對教學實作做自我性或社會性反思，再產生新的知識或信念，如此循環不已，教師個人的教學知能因而不斷成長。相對地，教師個人的知識、信念或教學亦會影響合作成長小組其他成員的知識、信念或教學。因此，合作成長小組成員透過彼此的互動，帶動彼此的成長。另一為數學教師的學習模型。數學教師依據數學教學知識或信念，透過社會性或自我性反思形成教學目標。根據教學目標產生教學行動的規則。其行動中的概念主要來自於數學教學知識或信念，據此對教學的情況進行分類和選擇有關教學的資訊。依據教學行動中的定理，從教學可得的相關資訊推論適當的教學目標及規則。由於教學知識或信念的改變，透過反思引動教學實作認知的改變。反之，教學實作認知的改變，透過反思引動教學知識或信念的改變。不同學習類型教師知、思、行三者間之不同互動模式，導致數學教學知能的不同成長結果。三者間越頻繁的互動，越能帶動教學知能的成長。
就理論的貢獻而言，由數學教師的學習模型可以瞭解不同學習類型教師的成長模式及促進其專業成長的策略。由數學教師個人與合作成長小組的互動成長模型可以瞭解合作成長小組互動成長的機制及促進互動成長的可行途徑。就教師教育的意涵而言，此二模型有助於在職教師專業發展計畫的規劃及實施，設計符合參與者學習特質的專業發展活動。就方法論而言，目前缺乏分析教師互動的有效工具。本研究質、量並重的創新分析方法可成為分析教師互動的可行工具。

The strategy to promote participant teachers’ professional development of mathematics teaching was through the interactions of the co-development group. The purpose of this study was to explore how and why participant teachers’ mathematics teaching competence and reflection grew as a result of his interactions with the co-development group. In the end, we were finally able to create a mathematical teacher learning model.
This is an interpretive study. The co-development group consisted of an investigator and three grade 1 teachers of a public elementary school in Taipei city. This co-operative intervention research lasted for one academic year. There were two reasons for choosing these three teachers as our subjects. The first reason was that these teachers were motivated to learn and taught the same grade in the same school. The second reason was that they had different teaching beliefs, experiences and personalities. The researcher played three roles: participant observer, facilitator and investigator in the process of this study. The group had regular weekly meetings to discuss their teaching. The activities that were adjusted were based on participants’ willingness, needs and collective goals. The main activities engaged in the study were analyzing patterns of students’ solutions and discussing cases of mathematics teaching, individual problems while teaching and literature.
The data collected for this study included classroom observations video-taped and audio-taped, group discussions audio-taped, teachers’ worksheets, teachers’ reflective journals, e-mail, documents of group discussions, analyzing patterns of students’ solutions, lesson plans, action research reports, students’ worksheets, the results of students’ tests and research journals. Participant teachers, two intern teachers, students and administrators were interviewed. All video-taped and audio-taped data were transcribed verbatim. The data were analyzed by using Cobb and Whitenack’s (1996) methodological approach which can be used to analyze large sets of qualitative data. The interactions of the group were analyzed quantitatively and qualitatively. The number of interactions and its percentage were calculated and visualized according to the non-equivalent events apparently reflecting the relationship between psychological processes and social processes. The teachers’ mathematics teaching competence and reflection were respectively analyzed according to Franke, Carpenter, Levi, and Fennema’s (2001) ‘Levels of Engagement with Children’s Mathematical Thinking’ and Ward and McCotter’s (2004) ‘Reflection rubric’. Multiple triangulation on the source, method, time and analyst were used to validate the data.
Five categories that distinguish the learning types of teachers stemmed from empirical data. Those categories were knowledge type, thinking model of learning to teach, the relationship between knowledge and action, teaching approach and belief about mathematics teaching. Four kinds of teacher’s learning types are distinguished by the categories and the frameworks. Those types are nave type, theoretical type, empirical type and practical type. The ultimate goal of a teacher is to become practical type.
Two models were induced from empirical data based on activity theory and cognitive theory of practice. The first model is called the Model of Interactions Between Individual Mathematics Teacher and Co-development Group. Making conflicts of cognition and encouraging experiments are the important mechanisms to facilitate the growth of a teacher. Co-development group facilitates individual teacher’s social reflection and instruction experiment. This is a cycle where an individual teacher produces new knowledge or belief by reflecting on his teaching internally or socially. Then he puts them into practice by reflecting internally or socially. As the cycle continues, his mathematics teaching competence will grow constantly. Relatively, an individual teacher’s knowledge, belief or teaching can also influence other participants in the same way . Thus, participants grow through interacting with each other. The second model is the Mathematical Teacher Learning Model. The mathematics teacher reflects his teaching internally or socially according to his knowledge or belief as a result of forming his teaching goal. His rules of action emerge based on his teaching goal. Concepts-in-action are mainly from his knowledge or belief. He categorizes and selects information about teaching by his concepts-in-action. He infers, from the available and relevant information about teaching, appropriate teaching goals and rules according to theorems-in-action. As his knowledge or belief changes, cognition of teaching practice changes via reflection, and vice versa. Different interactions between knowledge, reflection and action of different learning types result in different growth of mathematics teaching competence. The more knowledge, reflection and action interact, the greater the mathematics teaching competence.
As for contributions in theory, we can understand various learning models of different types and the use of different strategies to promote them to grow via the first model. We can understand the mechanisms and the viable ways to facilitate group development via the second model. As for teacher education implications, the models contribute to the plan and implementation of inservice professional development activities for teachers as it is designed to fit participants’ different learning types. As for methodology, a valid tool for analyzing teachers’ interactions is lacking. Therefore, the innovative method of this study can be a valid one.

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