This study was conducted to investigate how presenting two mathematical proofs of the Pythagorean theorem affects evaluating the beauty of the proofs. The authors conducted two experiments, each consisting of two sessions: a lecture on mathematical proofs of the Pythagorean theorem and evaluations of the beauty of the proofs of the theorem. In Experiment 1, the subjects in Group D attended a lecture in which two proofs that most people view positively were explained and evaluated their beauty. The subjects in Group S attended a lecture in which one proof that most people view positively was explained and evaluated its beauty. The results demonstrated that the evaluation scores of the beauty in Group D were higher than those in Group S. In Experiment 2, the proofs explained differed from Experiment 1, although the lecture and evaluations were the same. In Group D, two different proofs were explained: most people viewed one proof positively but not the other. In Group S only one proof, which was not viewed positively, was explained. The results showed that the evaluation scores of the beauty in Group D were lower than those in Group S. The results in Experiments 1 and 2 suggest that the difference of presentation with two proofs of the Pythagorean theorem had two different effects: one is the “beauty raising effect” or the “beauty lowering effect” and the other is the “beauty-difference expanding effect”.

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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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A strategy of collaborative learning for teaching the Volume and Temperature to 4th graders was developed for each sub-category of motivation-enhancing structures (i.e., Task, Authority, Grouping, Evaluation) proposed by Maehr & Midgley(1991). The purpose of this study was to investigate how our teaching strategies affect the scientific understanding and learning behaviors among peers in the lessons. The analyses of the data included a quantitative analysis based on pre and post unit questionnaires, a descriptive analysis based on the concept of “thermal motions of the atoms,” and an interpretive analysis based on students’ dialogue and behaviors during each class process. These analyses indicated the following :1) our teaching strategies promoted the acquisition of scientific concepts (volume increases but mass does not change with increased temperature). 2) expectancy from others in social relationships was promoted by introducing function of teaching strategy’s element supported for participation and discussion, and evaluated reciprocally among peers based on individually.

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[in Japanese]

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