分数の乗法・除法に関する代数的推論の明確化 : 記号論的視座から

抄録

This research aims at clarifying the transitional process from elementary mathematics to secondary one, especially arithmetic to algebra, and from the viewpoint of Early Algebra, we focus on algebraic reasoning at elementary grades. However, it does not yet become clear enough. The purposes of this paper, therefore, are to clarify algebraic reasoning through the semiotic analysis of classes of the multiplication and division with fractions, obtain the implications to promote algebraic reasoning, and obtain the implications for the classes. The results are followings. (1) Algebraic reasoning was identified by the semiotic analysis of the classes from the viewpoint of generalization and justification. The characteristic of the reasoning depends on deductive reasoning grounded on a property of numbers and operations or a pictorial expression. (2) From the viewpoint of justification, we regard deduction using a model and a specific case as deductive reasoning, in the case of deduction using a model, it is important that the model is made a tool, in the case of deduction using a specific case, it is important to examine whether become the generic example or the representative special case, and to draw various deductive explanations. (3) About the implications for the classes, at the time of classes of the multiplication at the second grade and the unifying partitive division with quotative division at the third grade, it is important to make students understand that the relation of multiplication to division is reverse operations from concrete operations. At the time of classes of the division with fractions, it is natural to understand the method of calculation from a property of operations.