Ed Dubinsky

Reflective Abstraction in Advanced Mathematical Thinking

Our purpose in this chapter is to propose that the concept of reflective abstraction can be a powerful tool in the study of advanced mathematical thinking, that it can provide a theoretical basis that supports and contributes to our understanding of what this thinking is and how we can help students develop the ability to engage in it. To make such a case completely, it would be necessary to do at least several things:

Development of the Process Conception of Function.

Our goal in this paper is to make two points. First, college students, even those who have taken a fair number of mathematics courses, do not have much of an understanding of the function concept; and second, an epistemological theory we have been developing points to an instructional treatment, using computers, that results in substantial improvements for many students. They seem to develop a process conception of function and are able to use it to do mathematics. After an introductory section we outline, in Section 2, our theoretical epistemology in general and indicate how it applies to the function concept in particular. In Sections 3, 4, and 5 we provide specific details on this study and describe the development of the function concept that appeared to take place in the students that we are considering. In Section 6 we interpret the results and draw some conclusions.

Understanding the Limit Concept: Beginning with a Coordinated Process Scheme.

Many authors have provided evidence for what appears to be common knowledge among mathematics teachers: The limit concept presents major difficulties for most students and they have very little success in understanding this important mathematical idea. We believe that a program of research into how people learn such a topic can point to pedagogical strategies that can help improve this situation. This paper is an attempt to contribute to such a program. Specifically, our goal in this report is to apply our theoretical perspective, our own mathematical knowledge, and our analyses of observations of students studying limits to do two things. First, we will reinterpret some points in the literature and second, we will move forward on developing a description, or genetic decomposition, of how the limit concept can be learned. In discussing the literature, we will suggest a new variation of a dichotomy, considered by various authors, between dynamic or process conceptions of limits and static or formal conceptions. We will also propose some explanations of why these conceptions are so difficult for students to construct. In describing the evolution of a genetic decomposition for the limit concept, we will give examples of how we used our analysis of interviews of 25 students from a calculus course to make appropriate modifications.

APOS: A Constructivist Theory of Learning in Undergraduate Mathematics Education Research

In this paper, we have mentioned six ways in which a theory can contribute to research and we suggest that this list can be used as criteria for evaluating a theory. We have described how one such perspective, APOS Theory, is being used in an organized way by members of RUMEC and others to conduct research and develop curriculum. We have shown how observing students’ success in making or not making mental constructions proposed by the theory and using such observations to analyze data can organize our thinking about learning mathematical concepts, provide explanations of student difficulties and predict success or failure in understanding a mathematical concept. There is a wide range of mathematical concepts to which APOS Theory can and has been applied and this theory is used as a language for communication of ideas about learning. We have also seen how the theory is grounded in data, and has been used as a vehicle for building a community of researchers. Yet its use is not restricted to members of that community. Finally, we point to an annotated bibliography (McDonald, 2000), which presents further details about this theory and its use in research in undergraduate mathematics education.

Some Historical Issues and Paradoxes Regarding the Concept of Infinity: An Apos-Based Analysis: Part 1

This paper applies APOS Theory to suggest a new explanation of how people might think about the concept of infinity. We propose cognitive explanations, and in some cases resolutions, of various dichotomies, paradoxes, and mathematical problems involving the concept of infinity. These explanations are expressed in terms of the mental mechanisms of interiorization and encapsulation. Our purpose for providing a cognitive perspective is that issues involving the infinite have been and continue to be a source of interest, of controversy, and of student difficulty. We provide a cognitive analysis of these issues as a contribution to the discussion. In this paper, Part 1, we focus on dichotomies and paradoxes and, in Part 2, we will discuss the notion of an infinite process and certain mathematical issues related to the concept of infinity.

On Learning Fundamental Concepts of Group Theory.

The research reported in this paper explores the nature of student knowledge about group theory, and how an individual may develop an understanding of certain topics in this domain. As part of a long-term research and development project in learning and teaching undergraduate mathematics, this report is one of a series of papers on the abstract algebra component of that project.The observations discussed here were collected during a six-week summer workshop where 24 high school teachers took a course in Abstract Algebra as part of their work. By comparing written samples, and student interviews with our own theoretical analysis, we attempt to describe ways in which these individuals seemed to be approaching the concepts of group, subgroup, coset, normality, and quotient group. The general pattern of learning that we infer here illustrates an action-process-object-schema framework for addressing these specific group theory issues. We make here only some quite general observations about learning these specific topics, the complex nature of “understanding”, and the role of errors and misconceptions in light of an action-process-schema framework. Seen as research questions for further exploration, we expect these observations to inform our continuing investigations and those of other researchers.We end the paper with a brief discussion of some pedagogical suggestions arising out of our considerations. We defer, however, a full consideration of instructional strategies and their effects on learning these topics to some future time when more extensive research can provide a more solid foundation for the design of specific pedagogies.

Learning Abstract Algebra with ISETL

From the Publisher: Most students in abstract algebra classes have great difficulty making sense of what the instructor is saying. Moreover, this seems to remain true almost independently of the quality of the lecture. This book is based on the constructivist belief that, before students can make sense of any presentation of abstract mathematics, they need to be engaged in mental activities which will establish an experiential base for any future verbal explanations. No less, they need to have the opportunity to reflect on their activities. This approach is based on extensive theoretical and empirical studies as well as on the substantial experience of the authors in teaching abstract algebra. The main source of activities in this course is computer constructions, specifically, small programs written in the mathlike programming language ISETL; the main tool for reflection is work in teams of 2-4 students, where the activities are discussed and debated. Because of the similarity of ISETL expressions to standard written mathematics, there is very little programming overhead: learning to program is inseparable from learning the mathematics. Each topic is first introduced through computer activities, which are then followed by a text section and exercises. This text section is written in an informal, discursive style, closely relating definitions and proofs to the constructions in the activities. Notions such as cosets and quotient groups become much more meaningful to the students than when they are presented in a lecture.

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.

Constructive Aspects of Reflective Abstraction in Advanced Mathematics

The notion of reflective abstraction was introduced by Piaget and, over a period of many years in a number of works, he expanded and elaborated this concept. He considered it to be the driving force of the (re-)constructions involved in the passage through the stages of sensori-motor actions, semiotic representations, concrete operations, and formal operations (Beth & Piaget, 1966, p. 245). But he also felt that reflective abstraction was critical for the development of more advanced concepts in mathematics. In his viewpoint, new mathematical constructions proceed by reflective abstraction (Beth & Piaget, p. 205). Indeed, it was for him the mechanism by which all logicomathematical structures are constructed (Piaget, 1971, p. 342), and he felt that “it alone supports and animates the immense edifice of logicomathematical construction” (Piaget, 1980a, p. 92).

Advanced Mathematical Thinking.

In this article we propose the following definition for advanced mathematical thinking: Thinking that requires deductive and rigorous reasoning about mathematical notions that are not entirely accessible to us through our five senses. We argue that this definition is not necessarily tied to a particular kind of educational experience; nor is it tied to a particular level of mathematics. We also give examples to illustrate the distinction we make between advanced mathematical thinking and elementary mathematical thinking. In particular, we discuss which kind of thinking may be required depending on the size of a mathematical problem, including problems involving infinity, and the types of models that are available.