動態幾何軟體GSP支援下之數學探究

Abstract

傳統的高中數學課程，主要以紙筆為工具進行抽象思考與推理的訓練。資訊科技融入教學的實施，則希望創造一個多元且互動性高的教學環境，不但能改進老師的教學方法，增進學生的學習效果，也能培養學生主動探索的精神及解決問題的能力。 很多人學習數學的歷程，都是直接從教科書或老師身上獲得數學公式及定理，然後努力學習證明及解題的技巧，過程中總是缺少由問題出發的探索經驗。本研究透過動態幾何軟體GSP所提供的環境，引導學生從事數學探究的活動，讓學生有機會體驗數學的求知活動中，由問題出發到實驗、觀察、猜測與驗證的完整過程。 我們從高中數學教材中選取三個主題，作為研究的題材。第三章透過數學實驗探索兩個分別與距離及面積有關的極值問題，並且透過兩個動態「圖說證明」的例子，將抽象的正餘弦函數疊合的概念以具體的動態圖形呈現。第四章探索利用軌跡及參數式作出圓錐曲線圖形的方法，以及多項函數圖形的作法。第五章從幾何的觀點探索複數的幾何性質，包括複數平面、複數的四則運算及複數的n次方根，並舉出幾個例子，說明這些幾何性質如何應用在解決平面幾何的問題上。 數學教育應該提供機會，讓學生體驗數學知識創造的過程，並享受發現的樂趣。動態幾何軟體GSP具有可操作、互動性高及可模擬問題情境的特性，我們相信運用GSP在數學的教學與學習活動上，將有助於提高學生學習數學的意願，並激發學生研究數學的興趣。

Traditionally, senior-high mathematical courses make students go into training in their abstract thinking and reasoning by pens and papers. The application of information technology to teaching attempts to create a multiple and highly-interactive teaching environment. This environment can not only improve teachers’ instructive methods but also cultivate students’ spirits of active exploring and abilities of solving problems. In the process of learning mathematics, many people directly acquire mathematical formulas and theorems from textbooks or teachers, and then strive to gain the skills in proving and problem solving. In this way, they always lack the exploring experiences triggered by problems. In the environment of GSP, this study aims to lead students to explore mathematics. In mathematical acquisition, students have better chances to experience the complete procedures for experimenting, observing, conjecturing and verifying, all initiated by problems. From senior-high mathematical materials, three subjects are selected as the topic in this study. In Chapter 3, two extremum problems are explored, related to distance and area by mathematical experiments. With two examples of dynamic “proof without words”, the abstract concept of the superposition of sine and cosine is demonstrated in concrete dynamic graphs. Chapter 4 probes the approaches to draw out the graphs of conic sections with locus and parametric equation as well as the approaches to create the graphs of polynomials. In Chapter 5, in geometric perspectives, the geometric properties of complex numbers are explored, including complex plane, four arithmetic operations of complex numbers and nth root of a complex number. It is also exemplified how these geometric properties are applied to solving the problems of plane geometry. Mathematical education is supposed to offer opportunities for students to go through the creation of mathematical knowledge and enjoy the pleasure of discovery. In the environment of GSP, learners can operate, highly interact and simulate some situations in question. We believe applying GSP to teaching and learning activities in mathematics will contribute to boosting students’ motivation for learning mathematics and further inspire their interest in studying mathematics.