數學系學生無窮小概念研究

A Study on Conceptions about Infinitesimals of College Mathematics Majored Students

本研究參考2010年的一篇論文〈大學生非標準無窮概念，譯自Nonstandard Student Conceptions About infinitesimals〉，修改該研究的問卷題目作為研究工具，測驗了某大學數學系一、二年級學生，收集79份有效問卷，並選取6位進行半結構訪談。訪談後依個案問卷表現、訪談內容、以及研究札記進行分析。研究發現：雖然已經學過微積分的無窮小觀念，但是大部分學生仍然以類似萊布尼茲對無窮小的想法填答，無窮小迷思普遍存在，仍然使用直觀想法說明無窮小概念，靠直觀學習可能是助力也可能是阻力。雖然學生高中時都學過0.9＝1，但是大部分學生都認為它們不相同，仍然使用直觀回答問題。體會「認知衝突」並有動機澄清它是學習抽象概念的重要過程。學生的迷思有：「存在無窮大與無窮小數字」、「存在二個無窮接近的數字」、「無窮小平方後便更小」、「無窮大與無窮小可以被計算」。學生可以分類成「自明」、「衝突自清」、「衝突它清」、「衝突未清」、「未明未清」五類。最後，建議大學數學系老師能反思這個結果，教學中適當的讓學生產生認知衝突，再利用學生「自清」與「它清」，增加成功學習機會。

Based on a 2010 article ''Nonstandard Student Conceptions About Infinitesimals'', this study modifies the questionnaire of the study as a research tool to test the freshman and sophomore of math majored college students. There are 79 valid questionnaires collected, and six students for semi-structured interviews. We analyze the performance of the questionnaire, the content of the interview and the research notes to reach theorems. The study found that although the concept of infinitesimal calculus has been studied, most students still use intuition to fill in the infinitesimal concept and still respond to ideas like Leibniz’s thinking. Learning by intuition may be helpful or resistance. Although students have learned 0.9=1 in high school, most students believe that they are not the same and still intuitively answer the questions. Understanding ''cognitive conflict'' and motivation to clarify it is an important process of learning abstract concepts. The myth of students is that ''existence of infinity and infinitesimal numbers'', ''existence of two infinity closed numbers'', ''smaller after squaring an infinitesimal numbers'', and ''calculatability of infinity and infinitesimal numbers''. Students can be classified into five categories: ''self-evident'', ''self-evident in conflict'', ''othersevident in conflict'', ''non-evident in conflict'' and ''unspecified''. Finally, it is suggested that university mathematics teachers should reflect on this result and improve their teaching.