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

Abstract

In this article, we obtain a genetic decomposition of students’ progress in developing an understanding of the decimal 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.

Résumé: Dans cet article, nous proposons une analyse du processus de compréhension, chez les futurs enseignants, de la décimale et de sa relation avec le nombre 1. Ce processus apparaît valide pour un nombre élevé de participants, c’est pourquoi nous proposons d’ajouter une nouvelle étape à l’APOS, qui constituerait la première modification significative à cette théorie depuis sa formulation (voir Dubinsky et Lewin, 1986). Notre analyse comprend une méthodologie relativement objective et hautement efficace qui pourrait s’avérer utile dans d’autres travaux de recherche ainsi que pour l’évaluation de l’apprentissage des étudiants.

Notes

1. The students involved in this study were preservice elementary and middle school teachers.

2. We distinguish between concept, which refers to the understanding of a mathematical idea within the mathematical community, and conception, which refers to an individual's understanding of that idea.

3. We did not investigate schemas in this study.

4. This idea resembles what Pirie and Kieren (1989) refer to as “folding back.”

5. Piaget (1975, 1974/1976) used both the terms substage and level. In our analysis we use the term level.

6. refers to repeating zeroes, with 1 at the end.

7. By reaching a particular stage or level, we mean that the participant was categorized according to that stage or level based on evidence of progress toward that level or stage. We do not rule out the possibility of shifts in thinking, nor do we dismiss the tentative or fragile nature of that thinking.

8. When counts did not suffice, making interpretations necessary, the authors made individual determinations and resolved differences collaboratively.

9. Detailed lesson plans are available from the authors.