確率ミスコンセプションのコモグニション論的解釈 ― 小数の法則に焦点をあてて ―

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  • Commognitive Interpretation of Probabilistic Misconception: Focusing on the Law of Small Numbers
  • カクリツ ミスコンセプション ノ コモグニションロンテキ カイシャク : ショウスウ ノ ホウソク ニ ショウテン オ アテテ

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<p>　　The purpose of this paper is to identify a structure of probabilistic misconception. Many people mistakenly assert that the Law of Large Numbers might be applied to small samples as well. Traditionally, in terms of misconceptions, this problematic situation is investigated through the Law of Small Numbers. This paper gives a new interpretation to this phenomenon from the perspective of commognitive conflicts. Consequently, it is shown that the Law of Small Numbers results from two metarules (namely the compensating rule and the generalizing one).</p><p>　　In this paper, a Triangle of Discourse (Figure 1) is constructed and used as a commognitivist framework for interpreting discourses. The triangle of discourse is roughly a commognized version of semiotic triangles on meaning. Then, the simplest commognitive conflicts are interpreted by two Triangles because commognitive conflicts arise between at least two discourses. Thus, a Dual Triangle of Commognitive Conflict (Figure 2) is constructed and used as a framework for interpreting commognitive conflicts. ST, RT, and MT are a learned discourse in a context (the subindex T indicates teachers), whereas SL, RL, and ML are a learners’ discourse in the context (the subindex L indicates learners).</p><p>　　Two types of misconceptions are distinguishable within the Law of Small Numbers when we take a detailed look at it. In the first type (LSN 1), “population / probability” as signifier is associated with “small sample” as realization. On the contrary, in the second type of misconception distinguishable within the Law of Small Numbers (LSN 2), “small sample” as signifier is associated with “population / probability” as realization. These two types of the Law of Small Numbers could be described by means of the dual triangles of commognitive conflict, as shown in Table 1. That is, the compensating and the generalizing rules are causes for the law of small number. The compensating rule is a metarule of routine procedure that we must compensate objects. The generalizing rule is a metarule of routine procedure that we must generalize objects. In learning probability, students should learn the new metarule, the swamping rule, which means that we must swamp objects.</p><p>Figure 1: Triangle of discourse</p><p>Figure 2: Dual triangles of commognitive conflict</p><p>Table 1: The two types of the Law of Small Numbers</p><p>LSN 1</p><p>ST RT MT</p><p>Population / Probability Large Sample Swamping Rule</p><p>SL RL ML</p><p>Population / Probability Small Sample Compensating Rule</p><p>LSN 2</p><p>ST RT MT</p><p>Large Sample Population / Probability Swamping Rule</p><p>SL RL ML</p><p>Small Sample Population / Probability Generalizing Rule</p>

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