Abstract

In Japan during the 1980s, there was an interesting debate about how to teach the area of a parallelogram effectively to primary school children. Yutaka Saeki criticized the standard method, which relies on a cut-and-paste procedure. He argued that the standard method inevitably failed to convince children because it does not provide any cogent reason for them to accept that the formula ‘base x height’ is indeed true. Saeki proposed his own method using a bundle of paper. This method, however convincing at first glance, was totally dismissed by Akira Nakagaki based on his orthodox scientific methodology. There emerged a lively debate between the two. By means of a reconstruction of this debate, this paper will show how the materiality of a thing can scaffold the process of gaining understanding of the concept of geometrical space. Although Saeki and Nakagaki were both unaware of the fact, the debate between them shows clearly that the convincing points of Saeki’s method rely on its material basis. The materiality of a thing (a bundle of paper in this case) can serve as a common basis, in which acquaintance in the context of everyday life is transgressed to the mathematical context. With this transgressive structure entailed in the materiality of a thing, children can be led to understand the spatial logic of the parallelogram, while they themselves make their own reasoning and judgement based on the context of their everyday lives.

摘要

在 1980 年代的日本，关于如何有效地向小学生教授平行四边形的面积发生了一场有趣的辩论。Yutaka Saeki 批评了依赖于剪切和粘贴程序的标准方法。他认为标准方法不可避免地无法说服儿童，因为它没有提供任何令人信服的理由让他们接受公式的底数x高度'的确如此。佐伯用一捆纸提出了自己的方法。这种方法虽然乍一看很有说服力，但基于他的正统科学方法论，被中垣明完全摒弃了。两人之间展开了热烈的辩论。通过对这场辩论的重构，本文将展示事物的物质性如何能够支撑获得对几何空间概念的理解的过程。虽然佐伯和中垣都没有意识到这一点，但他们之间的争论清楚地表明，佐伯方法的说服力在于它的物质基础。事物的物质性（在这种情况下是一捆纸）可以作为一个共同的基础，在这个基础上，日常生活背景下的熟人被超越了数学背景。