本研究欲探討數學系學生在學習基礎微積分時，對於函數極限的形式化定義是否會感到抽象？他們感受到的抽象性是有何內涵？這種抽象觀念從何而來？他們通常會如何處理及面對自己感到抽象的概念？學生們心中的抽象直覺與文獻中的抽象意涵是否具有相關性？或者要如何界定？

本研究採用自基礎微積分教科書上所節錄從口語化到形式化逐步調整的五項函數極限定義，以成對比較法作為問卷設計的主軸，讓受試者進行五項定義之間抽象程度的比較。量化分析的部分利用了多元尺度法的知覺圖瞭解可能影響受試者判斷抽象程度的重要因素；並利用Bradley-Terry模型探討細部不同群組之間的差異。最後再從有效問卷樣本的受試者中抽樣進行深入晤談，晤談時以五項定義之間的差異性比較的方式進行，為瞭解影響受試者判斷抽象程度的真正原因。結果發現，學生對於函數極限的形式化定義感到抽象的型態可簡單歸納為三種類型，第一類的學生將抽象視為難懂，但第二類與第三類的學生則與其將問卷中的五項函數極限定義轉譯為其心目中最能表達函數極限定義的不同方式有關；雖然學生心中的抽象直覺與文獻中的抽象意涵有所不同，但在學生進行成對比較與定義之間的差異性比較時，可以觀察到學生如何以文獻中的抽象思考方式，對五項函數極限定義產生數學性質的知識建構。

The main purpose of this study is to investigate the abstract perceptions of the students who major in Mathematics when they study the concepts of limit of the function in their calculus class.

We adopt five different descriptions of the definition of limit of the function from Spivak’s famous seminal textbook on Calculus. The five descriptions of definition are displayed from daily speaking language to formative mathematics language. The pair comparisons of these five descriptions of definition are designed in our questionnaire for the quantitative data analysis. We use the perceptual map of multi-dimensional scaling to analyze the possible abstract perception of students, and utilize Bradley-Terry model to cluster the different groups of students’ preference behaviors of abstract definitions. We also interview some students to look for the where their abstract perceptions come from.

At last, we conclude three types of students with different abstract perceptions, and found out their construction of reflective abstract as described in our literature review when students compare the difference between the five definitions of limit of function.

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