College students' intuitive understanding of the concept of limit and their level of reverse thinking
This thesis addresses the relationship of students’ intuitive understanding of the limit of a sequence to their reversibility, which is an ability to reverse the order between epsilon and N as required in the rigorous definition of limit. The subjects of this research were students who had not had any experience with rigorous proofs using the rigorous definition of a limit. Eleven students completed a series of 1-hour semi-structured, task-based interviews once a week for 5 weeks. Monotone bounded, unbounded, constant, oscillating convergent, or oscillating divergent sequences were tested. Students represented each sequence numerically as well as graphically in determining convergence of the sequence. Students also used tools, called epsilon-strips, specially developed for this study to explore the reverse relation in defining limits through hands-on activities with the strips. Finally, students were presented the following epsilon-strip definitions, and were asked to evaluate the propriety of the definitions as statements of the limit of a sequence. epsilon-strip Definition A: A certain value L is a limit of a sequence when infinitely many points on the graph of the sequence are covered by any epsilon strip as long as the epsilon strip covers L. epsilon-strip Definition B: A certain value L is a limit of a sequence when only finitely many points on the graph of the sequence are NOT covered by any epsilon strip as long as the epsilon strip covers L. It was found that students’ understanding of the definition of the limit of a sequence was associated with not only their conception of limit but also their level of reversibility. In addition, there was improvement in students’ reversibility and/or their conception of limit through the epsilon-strip activity, even though there was no procedure for indicating students’ errors, correcting students’ misconceptions about limit, or confirming the propriety of the epsilon-strip definitions to students during the interviews. This study implies that the epsilon-strip activity, combined with the various types of sequences used in this teaching experiment, is one instructional method for helping students develop and accommodate their conception of the limit of a sequence.
School:The Ohio State University
School Location:USA - Ohio
Source Type:Master's Thesis
Keywords:concept of limit sequences levels reversibility definition intuitive understanding
Date of Publication:01/01/2005