數學系學生對函數極限的錯誤認知與解題困境

Abstract

本研究的主要目的為探討數學系學生在微積分中對函數極限的錯誤認知與解題困境，以提供微積分教師在教學設計上的參考。本研究採用調查研究法，以自編的「函數極限測驗卷」為研究工具，對兩所國立大學修讀應用數學系微積分（一）的111位學生進行調查，分析學生的解題類型、錯誤認知與解題困境。研究結果發現：一、一半的學生無法正確描述函數極限的直觀意義與精確定義；二、學生未能理解函數極限的運算符號及使用極限計算法則的前提；三、學生無法釐清極限值、函數值與連續之間的關係；四、當函數含有根式、絕對值或有界的週期函數時，學生對計算函數的極限有困難。五、多數學生無法利用極限的精確定義證明線型函數的極限。最後依據研究結果，對微積分教學與研究提出一些建議。

This study investigated math students' misunderstandings and obstacles in learning limits of functions to enable calculus teachers to design their teaching plan accordingly. A self-designed test about the limits of functions was used as a tool in a survey to collect data from 111 students who were taking Calculus (I) at two public universities. The analysis of the students' problem-solving styles, misunderstandings, and difficulties pertaining to problem solving revealed the following findings: 1) Approximately half of the students could not correctly provide the intuitive and precise definitions of a limit; 2) the students could not understand the operational symbols of function limits and the basis for applying limit-related laws; 3) the students could not explain the relationship among limit, function value, and continuity; 4) the students experienced difficulties calculating the limit of a function when it contains radicals, absolute values, or bounded periodic functions; 5) most students could not apply the precise definition of a limit to prove the limit of a linear function. On the basis of these findings, we developed several recommendations for improving calculus teaching and research.