The Necessity of Demonstrating Theorem in the Triangle: Especially, ‘Proof as Means of Systematization’
2014 Volume 20 Issue 2 Pages 123-140
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2014 Volume 20 Issue 2 Pages 123-140
In our country, theorem in the triangle that says the three interior angles of a triangle add up to two right angles is often illustrated as a mathematical relationship, and many junior high school students do not understand why this needs to demonstrate. However, in a mathematics lesson of the second grade of a certain junior high school which we describe in this paper, a student whose name is Kan expressed the opinion which suggests the necessity of demonstrating theorem in the triangle.
The research purpose for us is to show clearly the meaning of this student’s opinion, and how this opinion was realized. We also aim to suggest the composition of the mathematics lesson which enables students to understand the meaning of demonstrating.
We analyzed the lesson using‘Didactical Situation Model (DSM)’ developed by Iguchi, Kuwano, and Iwasaki (2011), in order to achieve the above-mentioned research purpose. DSM is a framework based on mainly two ideas. One is a model developed by Mellin-Olsen (1991) identifying the level of knowledge control in a didactical situation. The other is the types of“MATOME” identified by Iwasaki and Steinbring (2009). And we made the viewpoint‘the functions of proof in mathematics’ identified by de Villiers (1990), especially‘proof as means of systematization’ when considering the lesson design which touches off the necessity of demonstrating theorem in the triangle.
The results of the analysis were summarized as follows:
(1) Kan’s opinion states that understanding the meaning of a student demonstrating theorem in the triangle, and it is characterized as a meaning of‘proof as means of systematization’.
(2) The following two are the important elements in the lesson for a student to express the opinion that Kan did: One is designing a lesson so that‘mathematics constituted locally’ is realized. Moreover, another is organizing a lesson so that‘an imperfect situation’ may occur.
