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Investigating flexibility, reversibility, and multiple representations in a calculus environment

Flanders, Steven (2015) Investigating flexibility, reversibility, and multiple representations in a calculus environment. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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This study investigates the development of flexibility and reversibility in a calculus environment that attends to linking multiple representations. Reversibility was studied through Krutetskii’s framework of reversibility of two-way processes and reversibility of the mental process in reasoning. The study was conducted over approximately four months in a high school calculus classroom in an urban school district in a mid-Atlantic state. Instruction attended to linking multiple representations whenever possible. Four types of data were collected: 1) a pre-test, 2) a post-test, 3) daily assessments, and 4) clinical interviews. Twenty-one students completed a pre-test and post-test that together assessed development of flexibility over the course of the study. They also completed daily assessments that were collected to provide evidence of the development of reversibility during the course of the study. Six students participated in four clinical interviews each, spread throughout the study. Inferential statistics were used to compare the results of the pre-test and post-test for significant differences and to determine significant differences in the presence of reversibility on the daily assessments over the course of the study. The clinical interviews were analyzed for evidence of students’ thought processes while solving reversible questions. Analysis revealed that over the course of the study, students demonstrated significant increases in both flexibility and reversibility. Two-way reversibility seemed to develop with relative ease for most students and often developed simultaneously with learning a forward process. Developing reversibility of the mental process in reasoning was difficult and tended to develop simultaneously with learning in a forward direction for students with high levels of flexibility. For students who did not develop reversibility simultaneously with forward learning, both two-way reversibility and reversibility of the mental process in reasoning were able to develop through multiple opportunities to solve reversible tasks of similar content. Analysis of the clinical interviews indicated that students typically followed a 4-step thought process when using reversibility to solve problems. Implications and limitations of the study and areas of further research were discussed.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Flanders, Stevenstf13@pitt.eduSTF130000-0003-3864-4955
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairAnsell, Ellenansell@pitt.eduANSELL
Committee MemberSmith, Margaretpegs@pitt.eduPEGS
Committee MemberMichael, Beverlybkm@pitt.eduBKM
Committee MemberGreeno, Jamesjimgrno@pitt.eduJIMGRNO
Committee MemberMunter, Charlescmunter@pitt.eduCMUNTER
Date: 27 January 2015
Date Type: Publication
Defense Date: 21 October 2014
Approval Date: 27 January 2015
Submission Date: 25 November 2014
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 496
Institution: University of Pittsburgh
Schools and Programs: School of Education > Instruction and Learning
Degree: EdD - Doctor of Education
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: calculus, flexibility, multiple representations, reversibility
Date Deposited: 27 Jan 2015 20:11
Last Modified: 15 Nov 2016 14:25


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