Abstract
The complexity of learning fractions on pupils' side has been underestimated. This has caused the traditional approach of teaching fractions to change into a more rigid and rule-oriented instruction. Thus, the teaching of fractions has become even more troublesome and complicated. We pay special attention to the fraction concepts that are taught preceding the teaching of fractions and the concepts that produce fractions. They are schematized based on two aspects: the different meanings of multiplication and Vergnaud's relational calculus on the additive structure. Consequently, we have formed three kinds of operational schema for the concepts. One of them is very useful for the analysis of the research tests and the classroom discussion in the third graders of Japan about the change of fraction concepts.
