Understanding the concepts of proportion and ratio constructed by two grade six students

The purpose of this study was to construct an understanding of two grade six students' proportional reasoning schemes. The data from the clinical interviews gives insight as to the importance of multiplicative thinking in proportional reasoning. Two mental operations, unitizing and iterating play an important role in student's use of multiplicative thinking in proportion tasks. Unitizing a composite unit and iterating it to its referent point enables one to preserve the invariance of a ratio. Proportions involved the coordination of two number sequences, keeping the ratio unit invariant under the iteration. In the iteration process, one needed to explicitly conceptualize the iteration action of the composite ratio unit to make sense of ratio problems and to have sufficient understanding of the meaning of multiplication and division and its relevance in the iteration process. One needed to have constructed multiplicative structures and iteration schemes in order to reason proportionally. This revised version was published online in July 2006 with corrections to the Cover Date.