Some Historical Issues and Paradoxes Regarding the Concept of Infinity: An Apos Analysis: Part 2

@article{Dubinsky2005SomeHI,
  title={Some Historical Issues and Paradoxes Regarding the Concept of Infinity: An Apos Analysis: Part 2},
  author={Ed Dubinsky and Kirk Weller and Michael A. McDonald and Anne Brown},
  journal={Educational Studies in Mathematics},
  year={2005},
  volume={60},
  pages={253-266}
}
This is Part 2 of a two-part study of how APOS theory may be used to provide cognitive explanations of how students and mathematicians might think about the concept of infinity. We discuss infinite processes, describe how the mental mechanisms of interiorization and encapsulation can be used to conceive of an infinite process as a completed totality, explain the relationship between infinite processes and the objects that may result from them, and apply our analyses to certain mathematical… 

Glimpses of Infinity: Intuitions, Paradoxes, and Cognitive Leaps

This dissertation examines undergraduate and graduate university students' emergent conceptions of mathematical infinity. In particular, my research focuses on identifying the cognitive leaps

Totality as a Possible New Stage and Levels in APOS Theory

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.

How to Act? A Question of Encapsulating Infinity

This article investigates some of the specific features involved in accommodating the idea of actual infinity as it appears in set theory. It focuses on the conceptions of two individuals with

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,

In search of $$\aleph _{0}$$ℵ0: how infinity can be created

A philosophical account of actual mathematical infinity that does not demand ontologically or epistemologically problematic assumptions is developed, based on a simple metaphor that is valid in terms of mathematical practice, as well as in line with empirical data on arithmetical cognition.

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

Being mathematical : an exploration of epistemological implications of embodied cognition

In this thesis I explore epistemological implications of embodied cognition in the hope of developing my apprehension of what it means to think mathematically. I allow my understanding of embodied

A Search for a Constructivist Approach for Understanding the Uncountable Set P(N)

This study considers the question of whether individuals build mental structures for the set ) (N P that give meaning to the phrase, "all subsets of N." The contributions of our research concerning

Analysis of factors influencing the understanding of the concept of infinity

The article discusses the understanding of infinity in children, teachers and primary teacher students. It focuses on a number of difficulties that people cope with when dealing with problems related
...

References

SHOWING 1-10 OF 37 REFERENCES

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

A Brief History of Infinity

In A Brief History of Infinity, the infinite in all its forms - viewed from the perspective of mathematicians, philosophers, and theologians - is explored, as Zellini strives to explain this

Infinity: A Cognitive Challenge

Dealing with infinity, an inherently abstract concept which defies concrete representation, involves distinct cognitive difficulties. I first review the research on the development of children's

Tacit Models and Infinity

The paper analyses several examples of tacit influences exerted by mental models on the interpretation of various mathematical concepts in the domain of actual infinity. The influences of the

Young Peoples' Ideas of Infinity

This paper considers views of infinity of young people prior to instruction in the methods mathematicians use in dealing with infinity. To avoid overlap with other papers in this special issue on

Letting the Intuitive bear on the Formal; A Didactical Approach for the Understanding of the Limit of a Sequence

This theoretical paper provides: (1) a presentation of some tasks that maybe regarded as typical sources for forming students' intuitions and first understandings about limiting processes of real

Infinity and the Mind: The Science and Philosophy of the Infinite

In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape, " where he explores infinity in all its forms: potential and actual, mathematical and

Finite and infinite sets: definitions and intuitions

The study explores the intuitive methods different-aged students use to determine whether a given set is finite or infinite; it also examines the relationship between these methods and the accepted,

Conflicts and Catastrophes in the Learning of Mathematics

The catastrophe theory model is explained in greater detail and the results of a test performed on first-year university students at Warwick University in collaboration with Rolph Schwarzenberger are reported.

A constructive response to `Where mathematics comes from'

Lakoff and Nuñez's bookWhere mathematics comes from: How the embodied mind brings mathematics into being (2000) provided many mathematics education researchers with a novel, and perhaps startling