Preservice Teachers’ Understandings of the Relation Between a Fraction or Integer and Its Decimal Expansion: Strength and Stability of Belief

Abstract

In an earlier study of preservice elementary and middle school teachers’ beliefs about repeating decimals, Weller, Arnon, and Dubinsky (2009) reported on a comparison of the mathematical performance of 77 preservice teachers who completed an APOS-based instructional unit with 127 preservice teachers who completed traditional instruction. The purpose of the current study, based on 47 interviews conducted 4 months after the instruction, during which time there was no further instruction on this topic, is to determine the strength and stability (over time) of the students’ beliefs, to uncover thinking that did not arise in the earlier study, and to see whether the interviews yield similar comparative results. The interviews did uncover changes in student thinking. The students who received the APOS-based instruction developed stronger and more stable (over time) beliefs that a repeating decimal is a number; a repeating decimal has a fraction or integer to which it corresponds; a repeating decimal in general equals its corresponding fraction or integer; and, in particular, In addition, a number of indices and categories were developed that may prove useful in other studies consisting of the comparison of interview and questionnaire data involving a large number of interview subjects.

Résumé

Dans une étude précédente sur les idées des futurs enseignants de niveau élémentaire ou moyen au sujet des décimales périodiques, Weller, Arnon et Dubinsky (2009) comparaient les performances en mathématiques de 77 futurs enseignants ayant terminé une unité pédagogique fondée sur le modèle APOS (action, processus, objet et schéma) à celles de 127 enseignants en formation qui venaient de terminer une formation traditionnelle. Le but de la présente étude, qui se fonde sur 47 entrevues menées quatre mois plus tard, après une période pendant laquelle ces mêmes notions n’avaient fait l’objet d’aucune formation, est d’analyser la force et la persistance (dans le temps) des idées des étudiants, de déterminer si les idées des étudiants ont subi des modifications par rapport aux données de la précédente étude, et de voir si les entrevues donnent des résultats comparables. Les entrevues indiquent effectivement que la pensée des étudiants a évolué. Les notions acquises par les étudiants ayant eu une formation fondée sur le modèle APOS sont plus fortement ancrées et persistantes, notamment pour ce qui est des idées suivantes: une décimale périodique est un nombre; à une décimale périodique correspond une fraction ou un entier relatif; une décimale périodique est généralement égale à la fraction ou à l’entier relatif qui lui correspond; et, en particulier, la décimale 0.999… = 1. De plus, nous avons mis au point certains index et catégories qui pourraient s’avérer utiles dans d’autres recherches qui comparent des données provenant d’entrevues ou de questionnaires destinés à un nombre élevé de répondants.

Notes

The written instrument appears in the Appendix.

We used pseudonyms for students referred to in this report.

About zero questions marked disbelief: If, for a given descriptor D#, one question was marked belief and zero questions marked disbelief, the student was categorized as tends toward belief. If two or more questions were marked belief and zero marked disbelief, the student was categorized as strong belief. A similar approach was used for the interview and stability categories that follow.

On some interview segments, more than one descriptor applied. In such cases, each descriptor was treated separately.

On some interview segments, more than one code was assigned. We counted each occurrence in our analysis and, when necessary, negotiated interpretations of ambiguous responses.

Four of the EXP students and three of the CTRL students did not comment about correspondence in either the written instrument or in the interview. Eight additional students, six from EXP and two from CTRL, addressed correspondence in the written instrument but limited their comments to equality in the interview. Consequently, percentages are based on the 17 EXP and 15 CTRL students who addressed correspondence in both.