教改爭議聲中，證明所為何事？

What for Proof vis-a-vis Education Reform Issues?

從1980（年）開始，解題、溝通與連結等數學能力，一直是數學教育努力的目標。而支撐這些能力的基本因子，就是數學論證能力。在本文第二節中，作者針對有關證明的研究作一些引述與評論，尤其提醒數學課程綱要的制/修訂者，如何『貼近』一些古代文本，以免陷入邏輯謬誤而不曾察覺。譬如說吧，美國加州公立學校數學架構中的幾何命題之邏輯順序安排，在歐幾里得《幾何原本》的脈絡下，就犯了循環謬誤。然後，在第三節中，作者進一步論述『視覺直觀』與『演繹論證』之間的折衷可能性，至於具體策略則可仿Freudenthal/Hanna & Jahnke所主張，設法從圍繞在幾何學中那些根本且有啟發性的應用面向，研擬出幾個『小理論』來。而在這些『小理論』的『局部組織』內，邏輯的嚴密性當然可以得到適當的照顧。再者，作者將在HPM的脈絡下，從貼近一些歷史經驗來尋找處理『證明』的出路，譬如在本文第四節中，我們所引述的Clairaut改編《幾何原本》時所注入的『發明的順序』之進路，乃至於劉徽的圓面積公式之『證明』等等，都說明了歷史經驗之可貴。因此，由本文論述來看，『證明』在數學教育過程中，不僅在於它的邏輯或論證『說服』，更重要的，應該是它對數學知識的『說明』功能，而這原本是數學教育工作者不應輕忽視之教育目標，在教改爭議聲中尤其更應有所堅持才是。

Mathematical capabilities like problem-solving, communication and connection have been primarily concerned by mathematics educators since 1980s. Underlying these capabilities there is a crucial denominator, namely, mathematical argumentation. In this article, the author will try to make a sketchy review on the related research publications. In the light of HPM, my theme will first be to remind designer of mathematics curriculum / editors of textbooks of the logical fallacy manifested with some conventional geometrical arguments. Take for example, logical order of geometrical propositions suggested by the Mathematics Framework for California Public Schools will apparently commit circular fallacy in the context of Euclid's Elements. Based on Freudenthal / Hanna & Jahnke's idea of "local organization", I will then go on to argue that a compromise between methodological visualization and logical rigor could be reached by teachers in the context of classroom practice. This may well explain how researches on HPM can benefit mathematics educators / teachers in a way that they can make proof more sense in their teaching as one can learn from Clairaut and Liu Hui. As a conclusion, I hope my argument in terms of the HPM has shown that proof should be presented not only to make convincing but more importantly, to explain mathematical knowledge.