Abstract
What we wish to show in this paper is to consider the difference between valuableness in algebraic symbols and that in geometrical figures in terms of the perspective of symbols as objects and the comparison between two teaching practices. Consideration on symbols as objects in algebraic signs was already reported referring to the class-analysis of learning through problem "Numbers on the Calendar" (1995, Iwasaki & Tagashira). This paper, therefore, is intended as a consideration on symbols as objects in geometrical figures referring to the class-analysis of learning through problem "The Sum of Five Angles in Pentagram". Development of geometrical figures as valuable corresponds to the semiosis from a drawing to form in geometry learning. The process of this symbols as objects is articulated as follows like problem, background, cognitive analysis, method and consideration, and conclusion in this order: (1) Problem It was almost impossible to identify geometrical figures as valuable "N" in symbols as objects concerning geometrical teaching clearly although it had been easy to specify algebraic symbols as valuable "n" in symbols as objects concerning algebra teaching. (2) Background of the Problem According to Thom, R., Euclidean geometry is a natural intermediate stage between common language and algebraic language (1973, p.207). Moreover according to Skemp,R.R., geometrical symbols and algebraic language contrast radically in mathematical representation. Natural language, therefore, has much more influence on geometrical figures than algebraic symbols. In Japanese, there are not distinction on singular or plural form of noun, and articles like as "the" and "a" either, which are deeply concerned with the establishment of logical quantifier. (3) Cognitive Analysis of Problem Students variously explained the reason on the proposition of the sum of five angles in pentagram in the class. Their explanations could be categorized four types. They are related to symbols as objects and proper to geometry learning. These four patterns are as follows: (1) by measurement (2) inductive explanation by using a demonstrative pronoun like as "this" and "that" (3) by expansion of special case like as regular pentagram (4) deductive explanation by using alphabetical symbols like as A,B,C,D, and E (4) Method and Consideration Semiotic trichotomy in modes of representation by Peirce corresponds to change of recognition as follows: (1) Icon as semiotic firstness is equivalent to an object of observation. A geometrical figure in measurement is a kind of icon. It, therefore, is undifferentiated from an object in thinking. This means a way of thinking depended on a concrete object. (2) Index as semiotic secondness is equivalent to the development of private schema. A geometrical figure referred with a demonstrative pronoun are a kind of index. It, therefore, is considered as representation of private schema. This means the possibility of inductive thinking. (3) Symbol as semiotic thirdness is equivalent to the development of collective schema. A geometrical figure referred with alphabetical signs are a kind of symbol. It, therefore, is considered as representation of collective schema. This means the deductive thinking rather than the extension of objects. (4) A geometrical figure in expansion of special case could be metonymy for that as a symbol. It is characterized as a transitional representation from private schema to collective schema. This implies the change of thinking way. (5) Conclusion It follows from what has been said in (1), (2), (3), and (4) that symbols as objects in geometry learning would be the fundamental base for generalization of method in thinking from induction to deduction. This is critically different from the generalization in algebra learning since symbols as objects there become the base for generalization of objects in thinking.
