Preservice Teachers' Understanding of the Relation Between a Fraction or Integer and Its Decimal Expansion

Abstract

Many studies establish that students at all levels, including preservice elementary and middle school teachers, have considerable difficulty understanding the relationship between a rational number (fraction or integer) and its decimal expansion(s), including the idea that 0.9 = 1. This article reports on the mathematical performance of preservice elementary and middle school teachers who completed a specially designed unit on repeating decimals that was based on APOS theory and implemented using the ACE teaching cycle. Students enrolled in a content course on number and operation at a large southern university participated in the study. Two sections received the experimental treatment, and three sections followed a traditional approach. The quantitative results suggest that the students who received the experimental instruction made considerable progress in their development of an understanding of the specific equality between 0.9 and 1 and the more general relation between a rational number and its decimal expansion(s). The students in the control group made substantially less progress.

Résumé

De nombreuses études ont montré que les étudiants de tous les niveaux, y compris les futurs enseignants au primaire et au premier cycle du secondaire, éprouvent de sérieuses difficultés à saisir la relation qui unit un nombre rationnel (une fraction ou un entier relatif) et son développement décimale, y compris par exemple la notion que 0.9 = 1. Cet article présente un compte-rendu de la performance mathématique d'un groupe de futurs enseignants au primaire et au premier cycle du secondaire qui venaient de terminer une unité pédagogique portant spécifiquement sur la répétition décimale, fondée sur la théorie APOS et mise en pratique grâce au Cycle d'enseignement de l’ACE. Les participants à la recherche étaient des étudiants inscrits à un cours théorique sur les nombres et les opérations dans une grande université du sud. Deux sous-groupes ont pris part au cours expérimental, et trois ont suivi une approche traditionnelle. Les résultats quantitatifs indiquent que les étudiants du groupe expérimental ont amélioré considérablement leur niveau de compréhension de l’égalité spécifique entre 0.9 et 1, de même que la relation plus générale qui existe entre un nombre rationnel et son développement décimale. Les progrès des étudiants qui faisaient partie du groupe contrôle ont été beaucoup moins importants.

Notes

1. In this report, the term fraction will be used to denote a rational number representation of the form a/b, a, b integers, b ≠ 0, and the phrase repeating decimal will refer to a repeating decimal representation of a rational number.

2. Although every integer is a fraction (a = a/1), we are finding that fraction and integer are very different for students, and we intend to deal with this in one of our future studies. Hence in this study, as well as in the Post-Instructional Written Instrument, we often refer to fraction or integer.

3. Let c₁, c₂, …, c_n, …, be an infinite sequence of integers, with 0 ≤ c_i ≤ 9. The number sup{0.c₁ c₂ … c_n | n = 1, 2, 3, …} is denoted by 0.c₁ c₂ … c_n … and is called an infinite decimal.

4. The algorithms presented in the bulleted items appear in a concise form for the purpose of this article. Course instructors did not use these “compressed” formulations in their class discussions.

5. In our discussion of the three sessions, we refer to strings. By this, we mean finite or infinite sequences of digits that correspond to the decimal expansion of a rational number.

6. For repeating decimals such as 0.3 and 0.35, where the cycle begins with the tenths place, the computer used the notation 0.3(3) and 0.3(53), respectively.

7. For readers wondering how a computer can store an infinite string, the answer is that the string is always a repeating decimal. The computer stores it in the form a.b(c). To perform any operation, the computer transforms the string to a fraction, performs the operation using standard fraction algorithms, and converts the result back to the string form.