培育自由想像的投影幾何課程

在108課綱的理念及數學領綱的目標中，並未提及自由想像，本文確認「自由想像」的數學素養在高中階段的重要性，並肯定以「投影幾何課程」來培育的價值。以史代納的教育美學為理論基礎，整理歸納出「以自由感知為基礎，經驗世界存有的美，以自由想像擴展視野，進而自由思考般的深入探索，輔以邏輯推理小心求證，做出自由的判斷」。分析投影幾何中從整體看部分的思路，引進無限遠點，將所有的關係一般化，讓圖形的形變過程「連續」起來，創造出整體圖像；介紹對偶關係，看見轉換立基點，又能符合邏輯推理的全新幾何風貌。
投影幾何課程的目的是透過精準繪圖，覺察自身感受，並運用邏輯推理得出無限及對偶的概念，訓練自由想像的能力，讓視野更開闊，豐富更多可能性，以自由感知與自由思考面對問題，用更有創意的全新方法來解決問題；並能懂得換位思考，透過自己的思辨，用不同的角度看事情，以多元觀點看待各領域的內涵。重視學生聽見內在真實聲音、自由判斷、批判性思考及美感教育四個部份，在培養學生邏輯推理的過程中，同時重視自由想像的重要性，產生驚奇的美感經驗。

In the 108 Curriculum Guidelines and the purpose of the Mathematics Curriculum, free imagination is not mentioned. This article confirms the importance of the mathematical literacy of ''free imagination'' at the high school level and affirms the value of cultivating it through a ''projective geometry course.'' Based on Steiner's educational aesthetics, the study proposes ''founded on free perception, experiencing the beauty of the world, expanding horizons through free imagination, and further exploring through free thinking, supplemented by careful verification through logical reasoning, making free judgments. It analyzes the approach of viewing parts from the whole in projective geometry, introducing the concept of infinity points to generalize all relationships, making the process of shape transformation ''continuous,'' thereby creating an overall image. It also introduces the concept of dual relationships, showing how changing the foundational perspective can reveal a new geometric outlook that aligns with logical reasoning.
The purpose of the projective geometry course is to cultivate the ability of free imagination through precise drawing, awareness of one's own perceptions, and logical reasoning to derive the concepts of infinity and duality. This training broadens perspectives and enriches possibilities, enabling students to face problems with free perception and free thinking, solving issues creatively and innovatively. It also teaches them to understand and practice empathy, looking at things from different angles and considering diverse perspectives across various fields. Emphasizing four aspects—hearing one's inner true voice, free judgment, critical thinking, and aesthetic education—this approach values the importance of free imagination in the process of developing students' logical reasoning. It aims to create an astonishing experience of beauty through the integration of logical reasoning and free imagination.