Asuman Oktaç

Development of students' understanding of cosets, normality, and quotient groups

This paper reports on a continuing development of an abstract algebra course that was first implemented in the summer of 1990. This course was designed to address discrepancies between how students learn and how they were traditionally being taught. Based on results from the first implementation, pedagogical changes were made, including increased computer programming activities and other exercises which were designed to give the students the opportunity to build experience to draw on in order to construct understanding of the topics in class. A second experimental course was run. To assess the impact of these methods, and to continue to better understand how students go about learning, test results from the latter class and interviews with students from both experimental courses and a lecture-based class were analyzed. The students in the second experimental course demonstrated a deep understanding of the title concepts, especially cosets and normality. We discuss the details of the revised experimental course; the epistemological theory behind its design; and the framework used to analyze the results. We demonstrate through examples from interviews and test results the applicability of this analysis to the data, and the strides made by the students in comparison with the students from the lecture-based course, and with the students from the first experimental course. We hope to illustrate difficulties students face in learning abstract algebra, and to discuss instructional strategies to help students overcome these difficulties.

Conversion, Change, Transition… in Research About Analysis

Research about analysis education is abundant, starting from the end of secondary school through the university level. This mathematical field, with its new objects, techniques, problems and queries, is indeed a source of difficulties for students. This chapter is concerned with change of all kinds: point of view, setting, register, stage, … related to work in, and comprehension of, analysis. It focuses its attention on emblematic and central notions of analysis: real numbers, limits, functions, derivation and integration.

Equity Issues Concerning Gifted Children in Mathematics: A perspective from Mexico

Traditionally, equity agendas in research have not involved the education of gifted children, in general, and attention to mathematically gifted children or youth, in particular. We consider that the equity issue concerning this population can be discussed at two levels: from an individual point of view taking into account dissatisfaction, loss talent and unmet needs, and from a societal point of view as loss of human resources and socio-economic gap between developed and developing countries. Extracts from an interview with the director of a program for talented children in Mexico City provide a context to emphasize the need to understand the specifics of each cultural and political system in order to deal with the equity issue and in order to avoid mistakes of “inequity within equity.” Special activity design for this population can be a means to attend the intellectual needs of gifted students.

Conceptions About System of Linear Equations and Solution

Understanding of systems of linear equations permeates in the study of several topics of importance in linear algebra, such as rank, range, linear independence/dependence, linear transformations, characteristic values and vectors. After giving an overview of the literature on the teaching and learning of systems of linear equations, research results on student difficulties at different school and university levels are presented, establishing relationships with the way this topic is taught. The conceptions that students develop about ‘system’ and ‘solution’ are discussed in synthetic-geometric and analytic contexts in two and three dimensional spaces. Based on these observations, some pedagogical suggestions about planning instruction on this topic are offered. Although the findings reported in this chapter correspond to research undertaken in Mexico and Uruguay, they might be reflecting a more general phenomenon related to conceptions that students develop in relation with systems of linear equations and their solutions.

Understanding and Visualizing Linear Transformations

The aim of this chapter is to give an overview of the research that we have been conducting in our research group in Mexico about the linear transformation concept, focusing on difficulties associated with its learning, intuitive mental models that students may develop in relation with it, an outline of a genetic decomposition that describes a possible way in which this concept can be constructed, problems that students may experience with regard to registers of representation, and the role that dynamic geometry environments might play in interpreting its effects. Preliminary results from an ongoing study about what it means to visualize the process of a linear transformation are reported. A literature review that directly relates to the content of this chapter as well as directions for future research and didactical suggestions are provided.

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.