電腦實作環境輔助多變數微積分的教與學研究

Abstract

本計畫旨在探查大學生在電腦實作環境中學習多變數的可微性其心路歷程與學習行為特徵。本研究擬採教學實驗研究法，欲利用發展中的電腦Java 3D教學模組讓可微性的相關概念視覺化，搭配工作單作為學習鷹架，引導學生基於其原有的切平面概念，探詢、聯繫以理解可微概念的幾何和代數表徵之間的關連。先前的研究讓我們瞭解視覺化電腦模組的探究活動大大有助於學生將代數表徵與圖像表徵作連結，建構可微性的意義，使得抽象的數學概念得以親近(tangible)，學生有踏實感，學習自信得以產生。但是我們對於學生的迷思概念與成因的理解仍顯不足，在這個研究計畫中我們想進一步探詢不同學習背景的學生，如大一初學微積分的學生和數學系修習高微的大二生，由於學習經驗與對可微性的認知層次的不同，他們在電腦實作環境的活動中所表現的對可微性的理解有何相似或差異性？電腦如何與其學習的心路歷程緊密交織，建立或修正可微性的理解？本研究希冀能對多變數函數的微積分教學提出一些實質上的建議。

The objective of this study is to investigate the behavior and thinking process as the students learning the differentiability of multivariable functions with computer. We will use the Java 3D interactive modules had been developed to yield the learning environment that provides opportunities for students to manipulate the mathematics materials represented in visual fashion. The teaching experiment methodology is employed. We designed problem-based worksheets as scaffoldings for learning the differentiability of multivariable functions, by helping students explore and connect their spontaneous concept of differentiability and tangent plane. From previous we had found that students can learn this topic meaningfully and confidently as they reasoned in visual way and built the connection between the graphic representation and their algebraic expressions of concepts. Students could aware and clarify their misconceptions through this learning environment. But the misconceptions and their cause we didn’t clear yet. In this project, we want to investigate along this line deeper and elaborative. For different level students, such as the calculus students and the advance students, how they could construct their understanding on the differentiability of multivariable functions with computer？And what is the difference or similarity of their learning behavior and misconceptions in such learning environment？This project will make some suggestions for teaching the differentiability of multivariable functions with computer.