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【繁體】數學也可以這樣學

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KingStone

【繁體】數學也可以這樣學

Name: 【繁體】數學也可以這樣學
Brand: KingStone
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一沙一世界，一花一天堂
飄落的雪花是幾何；太陽月亮是週期；葉子的節點是數列
換個方式學數學，你將發現自然的美麗及宇宙的秩序

華德福式自然學習法，超過200幅彩色圖表

本書是一位任教於華德福教育體系的教師，針對七、八年級學生所發展的教程，廣獲推介引用。藉由大量圖片與作品，引導學生認識大自然、空間以及時間裡的數學。主題包括：幾何學、畢達哥拉斯及數目、柏拉圖多面體、節奏與循環。

華德福的教育方式強調學習與經驗的連結。對教師和家長而言，點燃孩子的學習熱情更勝於填鴨教學。對學生而言，概念與觀察的結合會帶來驚喜與啟蒙。數學不只是計算與公式，更是探索、興趣與應用，也是一項重要生活技能。

推薦序
譯者序
導言

第一章 大自然中的數學
．技巧的複習與回顧．圓的形式．六邊形的形式．螺線的形式
．阿基米德螺線．等角螺線．斐波那契數及其數列．斐波那契螺線
．，斐波那契以及「黃金切割」．1.618 或 0.618？

第二章 畢達哥拉斯與數目
．為何畢達哥拉斯？．數目．質性的數目．各種數目系統
．十進位數目，指數寫法（長式）和我們普遍的簡寫形式
．長式和簡式寫法．二進位數．度量．距離與角．角的度量
．熟悉的度量工具．數目的種類．質數和伊拉托森尼斯篩子
．質數的篩子．畢氏三數組．畢氏定理．演示．婆什迦羅的證明

第三章 柏拉圖立體
．柏拉圖立體．歷史上的柏拉圖立體．平面圖形
．三種特殊的直角三角形．正立方體摺紙．三種三角形的細節
．碗和馬鞍．葉面及其孔洞和皺摺．中心點與外圍．四面體
．正四面體在哪裡？．正八面體．正八面體展開圖．正八面體實例
．正六面體（或正立方體）．正六面體展開圖．正六面體實例
．交錯穿插的正立方體和正八面體．正二十面體與正十二面體
．正二十面體展開圖．正二十面體的黃金分割結構．正十二面體
．再談黃金矩形．正十二面體展開圖．歐幾里得《幾何原本》第十三冊
．歐拉法則．學生作品

第四章 節奏與週期
．旋轉、節奏與週期．時間．輪子．圓和直徑
．圓周與直徑．阿基米德應用多邊形的進路．用正八邊形來計算
．的命名．的遞增精確度．圓周．微小、中等及巨大的尺寸
．圓形．白天、夜晚及內布拉星象盤．奠基於哥白尼的當代基本圖像
．季節．地球繞著太陽的橢圓路徑．克卜勒的行星運動定律
．大的和小的連結．人類和宇宙的節奏

謝詞
參考文獻
索引

內容試閱

導論

一篇發表於《雪梨晨間先鋒報》（Sydney Morning Herald）的文章（2001年12月20日），引述了服務於家長所經營的教會學校的梅卡飛（John Metcalfe）所說的一段話：「孩子被教導說，數學是一種描述世界的語言――上帝所創造的一種語言……」

這也是我多年來的感受，而且，只要嚴肅以對，我認為這是可以走得非常遠的一條進路。這條進路認為自然這本書（book of nature）有個祕密等著被揭露，還有，我們這個世界遠遠不只是一個長程的、機率般的偶然遇合，也不是透過各種令人不安的推斷過程而可以加以計算的，沒有任何具實務經驗的工程師會夢想可以這麼做。當然，吾人可以提出更多觀點，畢竟任#唯物論式#（materialistic）的科學壟斷了客觀性，是完全沒有道理的。

唯名論或是諸神的語言

有一種觀點主張，數學的世界是一些便利的理念（idea）的一種#唯名論式#（nominalistic）且抽象的集合體。這些理念本身少有（即使有）真實意義，有的只是與理解「外在」（out there）世界有關的便利性與實用價值。這種觀點雖然已經被思慮周延的學者們所批評，不過數學是如此有用的這個事實，我們通常是視而不見。

另一種觀點認為，數學，在多元的意義上，是諸神的一種語言。可以說，我們的心智對數學與幾何概念的理解，是「讓這個世界發生」的那些作用力最後的殘餘。這種觀點並不是假定我們所擁有的思想是知識性的便利，只是心智的影子，而認為它是一種真正的通道，引領我們進一步了解大自然這個工作室（workshop of nature）。吾人可以相信不僅存有諸神，甚至有可能去理解祂們#如何#運作。這一直是我的態度，而且我發現經由大自然所展現的奇蹟，以及我們正在研究的這個主題之美，我更加確定此事。

對我來說，莎士比亞的「思想的蒼白投射」只適用於我們現在淺薄的智性，而非思想生命可以到達的最終境界――正如魯道夫‧史泰納（Rudolf Steiner）在他的《靈性活動的哲學》（Philosophy of Spiritual Activity）一書所指出的：「有了思維活動，我們已經掌握了靈性的一個小小的角落。」

無庸置疑，在這些角落中，還有各種變貌，而這整件事情可以無止境地辯論下去。儘管有著知識論上的精確，數學概念與細心觀察的現象之協合所帶來的驚喜，還是可以讓我們忙於探索、好奇，以及深感興趣――它們無疑也是重要的。本書包括我教授七、八年級學生的主要課程內容。每一個課程單元都以超過三週的時間完成，其中在我們的學校，澳洲的史泰納學校（Glenaeon Rudolf Steiner School），每天早上都有一個半小時的上課時間。

每位教師都以不同的方式教授這些課程內容，而其成果就學生、教師、地點及時間而言，都是獨一無二的。不過，對我而言，似乎存在著一條我們共同努力打造的「黃金線」。

對每一位老師來說，華德福課程提供了持續的挑戰。挑戰在於為學生發展逐年的課程，也透過它發展我們自己的專業。我常在想，如果我們無法以身作則，又要如何要求學生發展（興趣）、學習以及成長（身心）呢？如果我們做不到，學生如何做到？教師與學生之間，必須是一種等式關係。

這些內容是我們對數學主題的貢獻。

我也要感謝許多學生與友人，他們的作品供我作為書中舉例。我只能在此申謝。倘若我無法親自指明他們的貢獻，在此也要誠摯致歉。

更不用說，這些內容材料只是我個人的選擇，其他許多人會納入其他的選擇。

然而，這是我使用了二十多年的教學材料，也引起許多學生與同僚的興趣，光從我影印的份數就可以看得出來！

約翰・布雷克伍德

第一章大自然中的數學

從青春期開始，年輕人便有一種越來越強的需求，他們希望能夠連結他們對於這個世界的想法，以及他們實際經驗到的世界。數學，尤其是幾何學，可以在我們周遭許多大自然的奇蹟中被發現。發自我們內在深處的某些東西，呼應了外在於我們的現象。

如果在孩子的教育生活與人文覺察（humanities consciousness）的發長間，有一種對應關係，所反映的（西方）歷史時期，就是文藝復興的早期或文藝復興之前。這種連結生命這兩個面向的個人需求，在那個時代開始出現，也強烈展現在青少年學生身上。這種對自主思想的奮戰，在面對時代的主流信仰時，可謂步步荊棘，因為那時候的人認為如哥白尼、伽利略與克卜勒這些個人的研究成果，對他們的信仰充滿威脅。而在現在這個年代，年輕人的心智啟蒙也挑戰著我們！

此時此刻，我們要#記住#（re-member）：在學生的生活中，他們的作品、進路及探索，才開始要爆炸，如同文藝復興時期的世界震盪與重塑。而我們需要走出原地，才做得到。以下所呈現的一些教學主題概要，來自我多年前在史泰納學校教過的一個特別的主要課程，我試著涵蓋我認為屬於這個時代的內容。當然，還有其他許多內容可以納入，我並未假裝完備，這只是我在當時所提供的部分內容。

我按照當時授課的順序，選取一些典型的練習。有時候是課堂活動建議，有時候是練習的指引。

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