本研究意在探討一位三年級教師發展數學推理規範及學生數學證明表現的歷程。研究聚焦於一位多年參與數學課室討論之教師，其在三年級上、下學期共五個單元中，建立數學推理規範的情形；以及班上三年級學生在課堂中數學證明的發展為何。
研究者以Stylianides(2007)所提出的證明三元素為判斷學生基礎論證之證明的標準，並分析教師在不同單元中，使用不同證明任務以發展學生證明之情形，本研究之結果根據以下三方面進行說明：
教師在數學課室中建立數學推理規範，對於發展學生證明的影響為正面的，本研究發現數學推理規範的建立可以促進學生的證明思考。
在單一案例證明任務中，學生的證明表現產生三種類型的論證模式。類型一為基礎論證不合乎證明，經教師介入後論證也不合乎證明；類型二為基礎論證不合乎證明，經教師介入，經教師介入後論證合乎證明；類型三為學生之基礎論證合乎證明，經教師介入後，證明提升到更進階的證明。而教師對於這三種類型的學生之基礎論證，進行不同教學介入，以補足或提升學生的論證，成為合乎證明的進階論證。
在無限案例證明任務在三年級課程中較少出現，在僅有的案例中，分析學生的證明表現為類型一的論證模式，學生以操作性證明的方式進行證明，不合乎證明。
綜上所述，本研究發現教師建立數學推理規範能促進學生證明的發展，並且學生的論證表現是否能達到證明，與教師的教學介入有關。

This study is intended to explore a third-grade teacher to develop the process of mathematical reasoning norms and the performance of students' mathematical proof. The study focuses on the teaching of mathematics in a third-grade teacher and her students.
The researcher used the definition of proof proposed by Stylianides (2007) to judge student's basic argument is qualified to as proof or not, then analyzed the teacher use different proof tasks to develop student proofs in different units. The results of this study are described as followings:
Teachers establish mathematical reasoning norms in mathematics classrooms and impact on the development of students’ proof.
In a single case of task, there are three types of proving activities. In type 1, the basic argument did not qualify as proof and ensuing arguments were not qualified as proof. In type 2, the basic argument did not qualify as proof, but the ensuing argument were qualified as proof. In type 3, the basic argument was qualified as proof and ensuing argument as a more advanced proof. The basic arguments of three types after the instructional intervention could facilitate the development of a proof or the development of a more advanced proof.
In infinite case of task, the proof missions appear less frequently in third grade courses. In the only one case, the analysis of the students' proof is expressed as the type 1 argumentation model. The student proves in the form of empirical proof, which were not qualified as proof and ensuing arguments were also not qualified as proof.
In summary, this research suggests that teachers establish mathematical reasoning norms can promote the development of students’ proof. And whether the students' argumentation performance can achieve the proof is related to the teacher's teaching intervention.

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