Preservice Teachers’ Understanding of the Relation Between a Fraction or Integer and its Decimal Expansion: The Case of 0.9̄ and 1

In this article, we obtain a genetic decomposition of students’ progress in developing an understanding of the decimal 0.9̄ and its relation to 1. The genetic decomposition appears to be valid for a high percentage of the study participants and suggests the possibility of a new stage in APOS Theory that would be the first substantial change in the theory since its inception (Dubinsky & Lewin, 1986). Our analysis includes a relatively objective and highly efficient methodology that might be useful in other research and in assessment of student learning.