Convergence and the Cauchy Property of Sequences in the Setting of Actual Infinity

Traditional definitions, language, and visualizations of convergence and the Cauchy property of sequences convey a sense of the sequence as a potentially infinite process rather than an actually infinite object. This has a deep-rooted influence on how we think about and teach concepts on sequences,...

Full description

Saved in:
Bibliographic Details
Published in:PRIMUS : problems, resources, and issues in mathematics undergraduate studies resources, and issues in mathematics undergraduate studies, 2013-04, Vol.23 (5), p.441-458
Main Author: Shipman, Barbara A.
Format: Article
Language:English
Subjects:
Active Learning
Actual infinity
analysis
Calculus
Cauchy sequence
Class Activities
College Mathematics
College students
Convergence
Conveying
Definitions
Infinity
Instructional Materials
Learning Activities
limit
Mathematical analysis
Mathematical Concepts
Mathematical Logic
Mathematics Instruction
Metaphors
Pedagogy
potential infinity
sequence
Students
Teaching Methods
Thinking Skills
Undergraduate Study
Validity
Verbs
Visualization
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Traditional definitions, language, and visualizations of convergence and the Cauchy property of sequences convey a sense of the sequence as a potentially infinite process rather than an actually infinite object. This has a deep-rooted influence on how we think about and teach concepts on sequences, particularly in undergraduate calculus and analysis. After characterizing this point of view, this paper reformulates the definitions of convergence and the Cauchy property in the setting of actual infinity. This yields a conceptually streamlined approach and simple proofs of classic results on sequences. The paper also presents pedagogical metaphors that guide students in defining limit and the Cauchy property from the actually infinite standpoint.
ISSN:1051-1970
1935-4053
DOI:10.1080/10511970.2012.753963