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古代數學論爭能告訴我們什麼？日本江戶時代關流與最上流著作中體現的數學素養價值
What Can Pre-Modern Debates in Mathematics Tell Us? The Values of Mathematical Literacies Reflected in the Works of the Seki School and the Saijo School in Japan's Edo Period
摘要
本文嘗試從日本江戶時代關流與最上流之間發生的數學論爭中，找出其爭議焦點所體現的數學素養價值。本文所採用的素養觀點，主要來自左台益與李健恆總結之數學素養意涵。從他們的論述可推論出，數學素養不只是在真實世界使用數學的能力，傳統上在數學世界解決問題與價值判斷的能力也包含在內。從這樣的觀點出發，本研究的發現如下：首先，最上流追求精簡，直指題目的數學概念，而關流的問題有時使用生活實例，數值較繁瑣。從數學素養的角度來看，我們希望使用實例作為教學，但實例可能有複雜的數值，計算上造成學生較多負擔。如何設計問題數值不致影響學習過程，是現代教育工作者需要權衡之處。其次，小數值固然可能好計算，但情境如果來自真實世界，那麼誤差的問題就需要考慮，這裡體現的數學素養價值，就是數值的設計要考慮計算與取概數的過程，即處理與數字相關議題之素養。最後，數學素養包含在各種數學實踐中所需要的能力，也延伸至對數學方法的評估與判斷。在代數思考中，算則或證明在過程中需要設定幾個未知數或變數，需要看學習者的脈絡而定。使用變數較多的過程，得到的算式可能較為簡潔，而使用較少的變數，過程的算式可能會較為複雜。在教學上，我們需要理解學習者的脈絡來選擇問題內容，以及論證過程中變數或未知數使用的個數。簡而言之，從江戶時代數學論爭中，我們可以看到數學教育工作者需要：（1）在使用真實問題教學時考慮問題複雜度與教學目標之間的平衡；（2）使用真實數值的過程中帶領學習者考慮取概數對最終解答的影響；（3）進行代數推論時考慮學習者的脈絡平衡變數數量與算式複雜度。以上是本文的發現與對教學的建議。
並列摘要
This paper explores the values of mathematical literacies reflected during the mathematical debates between the Seki School and the Saijo School in Japan's Edo period. The framework of mathematical literacies adopted by this paper mainly comes from the viewpoints of Tai-Yih Tso and Kin Hang Lei. From their perspective, we can infer that mathematical literacies are not only the abilities to use mathematics in real life but also those of problem-solving and value judgements in the world of mathematics. From this viewpoint, the following are the findings. First, the Saijo School pursued conciseness, pointing directly towards the major mathematical concepts in questions, while the Seki School sometimes used real-life examples for teaching, but those examples might have complicated numerical values, resulting in more burdens for learners. How to design the numerical values to balance between reality and learning processes is an important question for educators to consider. Second, small numerical values might be easier to manipulate, but if the problem comes from the real world, then we also need to think about the issue of errors. The value of mathematical literacies reflected here is that the design of numerical values in problems needs to take into consideration the process of calculation and its errors, which is related to the literacy of handling issues of numbers. Last but not least, mathematical literacies include the abilities necessary in all mathematical practices, and by extension, they include the assessments and judgements of mathematical methods. During the process of algebraic thinking, how many unknowns or variables are needed in the process of an algorithm or a proof depends upon the context of the learner. The process with more variables might result in simpler equations / expressions, while using fewer variables may lead to more complicated equations/expressions. We need to adjust the problem and the number of variables according to the learner's context. In short, from the mathematical debates presented in this paper we can see that mathematics educators need to (1) balance between complexities and teaching goals when using real-world problems in teaching, (2) guide the learner to consider the influence of approximations on final solutions, and (3) consider the learner's context and balance between the number of variables and the complexities of algebraic expressions. These are our findings and suggestions for teaching.