本研究旨在探討國小教師之數學證明教學知識與課室數學推理規範對於學生數學證明活動表現之影響，本研究以Stylianides(2007)提出的數學證明三元素為判斷學生論證之證明的標準，並採用個案研究法，觀察一位長期進行證明教學的專家教師進行證明教學之情形，並分析專家教師在不同的證明任務下促進學生證明表現之情形以及建立的課室數學推理規範，再以訪談的方式分析專家教師運用的數學證明教學知識，本研究獲得以下結論：

一、在教師建立的數學推理規範有，分別為運用已知的數學知識作為說明的依據、理解不同解題背後的想法、證明使推理一般化以及利用多種表徵協助思維，此四種規範有效促進學生產生Stylianides(2007)提出的數學證明三元素，達到證明表現。
二、教師應具備的證明教學知識為兩種，第一種，了解數學證明三元素與其標準，符合Stylianides和Ball(2008)提出的證明教學知識中證明的邏輯語言知識；第二種，教師能針對學生在不同證明任務活動中學生的基礎論證中缺乏的元素進行有教的教學策略，符合Stylianides和Ball(2008)提出的證明教學知識中的證明情境知識。
三、在教師的證明教學知識與課室數學推理規範的影響下，大部分學生的論證能達到證明的標準。

最後根據研究結論，分別針對欲從事證明教學的教師與相關研究者，提出具體的建議以供參考。

This research aims to explore the impact of elementary teachers’ teaching knowledge of mathematics proof and classroom mathematics reasoning norms on the performance of students’ mathematical proof activities. This study adopted the three elements of mathematical proof proposed by Stylianides (2007) as the criteria for judging the proof of student’s argument. The research used the case study method and observed an expert teacher who has been teaching proof for years to conduct proof teaching. Then further analyzed the situation of the expert teacher promoting students’ proof performances under variate proof tasks, as well as the established classroom mathematical reasoning norms. Lastly, an interview was conducted to analyze the expert teacher’s teaching knowledge on mathematics proves. This research has the following conclusions:
I.The mathematics reasoning norms established by teachers include using known mathematical knowledge as the basis for the explanation, understanding the underlying ideas of different problems, proofs to generalize the reasoning, and the use of multiple representations to assist thinking. Such four norms effectively promote students to produce the three elements of mathematical proof proposed by Stylianides (2007) to achieve proof performance.
II.There are two kinds of proof teaching knowledge that teachers should possess. One is to understand the three elements of mathematical proof and its standard basis, which conforms to the logical language knowledge of proofs in the proof teaching knowledge proposed by Stylianides and Ball (2008). The other is the teacher can provide proper strategy based on the elements that students lack in the students' basic argumentation in different proof task activities, which is in line with the proof context knowledge in the proof teaching knowledge proposed by Stylianides and Ball (2008).
III.Under the influence of teachers’ teaching knowledge of mathematics proof and classroom mathematics reasoning norms, many students’ argumentations can reach the standard of proof.

Finally, according to the research conclusions, the study provided specific suggestions for teachers who engaged in proof teaching and related researchers for reference.

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