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By Terēze Budin̦a-Lazdin̦a

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Here too the translation follows Lam and Ang, 1992, p. 56). Whether or not this argument is convincing, one must ad­ mit that Liu Hui did tell us the reason why the algorithm ("Geng Xiang Jian Sun") worked successfully in simplify­ ing the fraction. Meanwhile, due to his argument that "the mutually subtracted numbers are all multiples of the 'Deng Shu"', one cannot help thinking of his structural concern in the context of algorithmic mathematics. In contrast, Euclid was primarily interested in finding a common measure (divisor) of two [natural] numbers.

We are instructed first to layout ~ ~ 7 7 7 7 7 7 91 42 42 35 28 21 14 7 then to divide the numerator and denominator by the "equal number" 7 in order to get the answer 7/13. According to Van der Waerden, since it is quite easy to get the an­ swer in this case, "the mention of the algorithm is not a logical or didactical necessity; it is just an addition by a systematically-minded teacher, who wanted to teach a never-failing method" (Van der Waerden, 1983, p. 38). Even so, an explanation of how the algorithm works made by Liu Hui in his commentary to the Jiu Zhang Suan Shu is equally impressive: To reduce by using the "Deng Shu" means to divide [the denominator and numerator] by it.

Then the question is not about the quantity of transmitted knowledge, but about its quality. Why did a certain concept arise? Under which historical conditions? The traditional idea that the development of mathematics is purely cumu­ lative is largely out-of-date. Teachers must remain aware of the inherent relativity of knowledge, and of the fact that, in the long run, providing students with an adequate view of how science builds up knowledge is more valuable than the mere acquisition of facts (von Glasersfeld, 1991).

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