Understanding “the Principle of Exchanging Variables” in a Formula of Intensive Quantity

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  • 内包量の公式における「変数の入れ替え原理」の理解
  • ナイホウリョウ ノ コウシキ ニ オケル 「 ヘンスウ ノ イレカエ ゲンリ 」 ノ リカイ

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Intensive quantities, such as speed and population density, among others, are generally represented as the quotient of two quantities. Even if the divisor and divided are exchanged, it is possible to compare the “strength” of an intensive quantity (a basic concept in elementary mathematics textbooks). In the present study, this concept was named, “the principle of exchanging variables.” It was examined from the following perspectives: (1) understanding arbitrariness (variables are arbitrarily assigned to the divisor and divided), (2) understanding reversal (when variables are exchanged, the magnitude order of the results is reversed), (3) understanding meaning (understanding what is calculated with the formula in which variables have been exchanged). In this study, the following intensive quantities were used: “the degree of congestion by people” and “the degree of profitableness of a part-time job.” In Study 1, understanding of arbitrariness and reversal was examined with university students (N = 110). In Study 2, understanding of meaning, arbitrariness, and reversal was examined with first-year junior high school students (N = 133). The results indicated that the percentage of participants understanding these concepts was very low in both studies. It is suggested that learners consider only the formulas taught at school (the number of people÷the area) and those used in daily life (wages÷time) as correct. The above results are discussed from the perspectives of “ossified view of formulas” and “knowledge operation” presented by kudo (2005).

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