Material basis of learning: From a debate on teaching the area of a parallelogram in 1980s Japan

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.

Keywords:

• Materiality
• paradox of learning
• area of parallelogram
• teaching media
• introductory understanding

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes on the article

The article was first read at the international conference ‘Meaning of Things in Educational Space’ held in March 2017 in Tokyo as an interim summary of the 3 years project ‘Things and Media in Educational Space’ supported by the Japan Society for the Promotion of Science (JSPS). Its Japanese translation was published 2018 in the final report of the project.

Notes

1 Wertheimer points out the same difficulty by reporting the reactions of 12-year-olds who were told of the procedure of “diminishing the disturbances by cutting horizontal rows with altitudes smaller than any assignable quantity, infinitesimally small. (…) After being told the whole story some were still puzzled, thought it ‘insincere’ even when shown that if the rows are shifted properly horizontally, the whole figure becomes more and more ‘similar’ to the rectangle.” (Wertheimer, 1959, p. 49)

2 According to the standard theorem of philosophy, the criteria as to be able to say “A knows q” are following:

i) q is truth (criterion of truth),

ii) A believes q (criterion of belief), and

iii) A has enough reasons for q (criterion of reason).

3 Saeki mentions in a later essay written with his colleagues that the understanding the students gained by the paper-bundle-method was “deeply and firmly rooted in their everyday intuition of manipulating concrete objects.” Children “may acquire, through various mediational tools, formal operational concepts which are strongly tied to (and therefore intuitively accepted as) the truth ‘afforded’ by the reality of objects in the world.” (Saeki et al., 1991: 241) Nevertheless, it remains unexplained how such a relationship to the “concrete objects” or the “reality of objects in the world” could lead to the desired understanding regarding the area of a parallelogram. Relying on the magic word “affordance” seems to me to be insufficient.

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Notes on contributors

Yasuo Imai

Yasuo Imai is professor of philosophy of education at the Faculty of Integrated Arts and Social Sciences of Japan Women’s University, Tokyo. He edited with Christoph Wulf Concepts of Aesthetic Education. Japanese and European Perspectives (2007). Besides the books on Walter Benjamin, media theory of education, and history of modern German educational thought, all written in Japanese, he also has published many papers in German and English, such as ‘Massenmedien und Bildung. Eine pädagogische Interpretation der Adorno-Benjamin-Kontroverse’ (1997), ‘Walter Benjamin and John Dewey: The Structure of Difference between Their Thoughts on Education’ (2003), ‘Why Does Language Matter to Education? A Comparison of Nietzschean and Wittgensteinian Views’ (2011).