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.