Kirk Weller

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

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 0.9̄ = 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 0.9̄ 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 progresst.RésuméDe 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 0.9̄ = 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 0.9̄ et 1, de mêe 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.

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

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, 0.9̄ = 1. 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.RésuméDans 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.

Preservice Teachers’ Understanding of the Relation Between a Fraction or Integer and its Decimal Expansion: The Case of 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.RésuméDans cet article, nous proposons une analyse du processus de compréhension, chez les futurs enseignants, de la décimale 0,9̄ 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.

Use of APOS Theory to Teach Mathematics at Elementary School

Throughout the first half of the 1990s, the mathematics team of the Center for Educational Technology, Tel-Aviv, Israel(CET), set out to revise the team’s existing materials for teaching mathematics in Israeli elementary schools (Grades 1–6, ages 6–12). One important aspect of the revision was to introduce the ideas of Piaget and APOS Theory into the teaching sequences. An area of particular interest was the teaching of fractions in grades 4 and 5.

The Teaching of Mathematics Using APOS Theory

This chapter is a discussion of the design and implementation of instruction using APOS Theory. For a particular mathematical concept, this typically begins with a genetic decomposition, a description of the mental constructions an individual might make in coming to understand the concept (see Chap. 4 for more details). Implementation is usually carried out using the ACE Teaching Cycle, an instructional approach that supports development of the mental constructions called for by the genetic decomposition.

Totality as a Possible New Stage and Levels in APOS Theory

The focus of this chapter is a discussion of the emergence of a possible new stage or structure and the use of levels in APOS Theory. The potential new stage, Totality, would lie between Process and Object. At this point, the status of Totality and the use of levels described in this chapter are no more than tentative because evidence for a separate stage and/or the need for levels arose out of just two studies: fractions (Arnon 1998) and an extended study of the infinite repeating decimal \( 0.\bar{9} \) and its relation to 1 (Weller et al. 2009, 2011; Dubinsky et al. in press). It remains for future research to determine if Totality exists as a separate stage, if levels are really needed in these contexts, and to explore what the mental mechanism(s) for constructing them might be. Research is also needed to determine the role of Totality and levels for other contexts, both those involving infinite processes and those involving finite processes. It seems clear that explicit pedagogical strategies are needed to help most students construct each of the stages in APOS Theory and that levels which describe the progressions from one stage to another may point to such strategies. Moreover, observation of levels may serve to help evaluate students’ progress in making those constructions.

Mental Structures and Mechanisms: APOS Theory and the Construction of Mathematical Knowledge

The focus of this chapter is a discussion of the characteristics of the mental structures that constitute APOS Theory, Action, Process, Object, and Schema, and the mechanisms, such as interiorization, encapsulation, coordination, reversal, de-encapsulation, thematization, and generalization, by which those mental structures are constructed.

From Piaget’s Theory to APOS Theory: Reflective Abstraction in Learning Mathematics and the Historical Development of APOS Theory

The aim of this chapter is to explain where APOS Theory came from and when it originated. A discussion of the main components of APOS Theory—the mental stages or structures of Action, Process, Object, and Schema and the mental mechanisms of interiorization, coordination, reversal, encapsulation, and thematization—points to when they first came on the scene and how their meanings developed. The published research of those involved in the development of APOS Theory, which includes some early colleagues and students of Dubinsky as well as those who were members of the Research in Undergraduate Mathematics Education Community (RUMEC), is described. The descriptions in this chapter are very brief and will be expanded in later chapters.

The APOS Paradigm for Research and Curriculum Development

In the Merriam-Webster online dictionary, the word paradigm is defined in the following way: “a philosophical and theoretical framework of a scientific school or discipline within which theories, laws, and generalizations and experiments performed in support of them are formulated; broadly: a philosophical or theoretical framework of any kind.” This definition reflects the contemporary meaning of the term coined by Kuhn (1962), who spoke of two characteristics of a “paradigm”: A theory powerful enough to “attract an enduring group of researchers” (p. 10) and to provide enough open ends to sustain the researchers with topics requiring further study. In light of these considerations, the overarching research stance linked to APOS Theory is referred to as a paradigm, since (1) it differs from most mathematics education research in its theoretical approach, methodology, and types of results offered; (2) it contains theoretical, methodological, and pedagogical components that are closely linked together; (3) it continues to attract researchers who find it useful to answer questions related to the learning of numerous mathematical concepts, and (4) it continues to supply open-ended questions to be resolved by the research community.

Schemas, Their Development and Interaction

APOS Theory has been successful in describing and predicting the types of mental structures students need to construct in order to learn abstract concepts. As new research is carried out and complex research projects are undertaken, it has become necessary to widen the scope of the theory. This has been achieved by expanding the researchers’ understanding of various theoretical constructs. Although there has been less research using these constructs, they already form part of the theory or are being tested in current research. One of these constructs is Schema; another is the mechanism of thematization and another, to be discussed in Chap. 8, is a possible new stage, Totality, between Process and Object.