本研究探討國小五年級學生面對數學非例行性問題的表現，分析數學探究活動實施前後，非形式推理數學內容面向、層級與歷程的差異與改變。研究方法為質性研究輔以量化分析，先施以前測，再進行三次數學探究活動，最後進行後測。參與者為1位教師和18位學生。藉由分析問卷、活動紀錄與晤談紀錄發現：經過數學探究活動後，數學內容面向增加「因數倍數面向」、「圖像聯想推理面向」及「數字關係推理面向」。因為學生推理時引用許多證據建立主張，並產生反駁，所以推理層級組成更加充分。最後，選擇6位學生進行個案研究，非形式推理歷程分為第一類逐漸變為複雜、第二類立場改變、第三類由複雜變為簡單後再變為更加複雜。學生透過含有討論與思考的活動才能有效增進推理的解題能力，而探究活動是其中一個適合讓學生增進推理能力的教學方式。

This study explored the performance of informal reasoning of mathematical non-routine problems and analyzes the differences and changes before and after the inquiry activities for the fifth-grade students. The research method included qualitative analysis and supplemented by quantitative analysis. The participants were 1 teacher and 18 students. Through the analysis of questionnaires, activity records and interview records, it is found that after the mathematical inquiry activities, the mathematical content is oriented to increase the "factor multiple orientation", "image association reasoning" and "digital relationship reasoning". Because the student reasoning uses many evidence to establish claims and generate rebuttals, the level of reasoning is more complete. Finally, we apply the case study on chosen 6 students. The process of informal reasoning is divided into the first class gradually becoming complex, the second class changing position, and the third class becoming complex from simple to simple. Students can effectively improve the problem-solving ability of reasoning through activities with discussion and thinking, and inquiry activity is one of the teaching methods suitable for students to improve their reasoning ability.