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Calculus III Students' Analytic and Visual Understanding of Surface Areas of Spheres, Cylinders, Pyramids and Prisms

Title: Calculus III Students' Analytic and Visual Understanding of Surface Areas of Spheres, Cylinders, Pyramids and Prisms.
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Name(s): Meadows, Yelena A., 1976-, author
Shaw, Kenneth, professor directing dissertation
Huckaba, Sam, outside committee member
Aspinwall, Leslie, committee member
Jakubowski, Elizabeth, committee member
School of Teacher Education, degree granting department
Florida State University, degree granting institution
Type of Resource: text
Genre: Text
Issuance: monographic
Date Issued: 2008
Publisher: Florida State University
Florida State University
Place of Publication: Tallahassee, Florida
Physical Form: computer
online resource
Extent: 1 online resource
Language(s): English
Abstract/Description: Calculus is often viewed as a gateway to more technical college majors, such as engineering, computer science and teaching of mathematics. The majority of existing research in college calculus classes has been focused on reports and studies that were conducted within the single-variate calculus content. Some researchers explain the high rate of students changing their majors from science-oriented to less mathematically rigorous majors due to students' struggles with passing multivariate content of the calculus college sequence (calculus I, II, and III). A qualitative case study of one section of a calculus III class was undertaken in order to obtain descriptive data on students' visual and analytical understanding of surface areas of familiar shapes of spheres, cylinders, prisms, and pyramids in the context of multivariate calculus. Specifically, this research focused on application of the surface area formula of surfaces described by a function of two variables. In the course of semester-long study, observed students divided into three distinct groups according to their mathematical visualization preference and mathematical accuracy characteristics. The three cases are: (1) students who prefer analytical method of solving mathematical problems with above average mathematical accuracy; (2) students who prefer visual method in solving mathematical problems with above average mathematical accuracy; and (3) students who prefer visual method of solving mathematical problems with below average mathematical accuracy. There was no group of students with below average mathematical accuracy preferring analytical methods of solving mathematical proeblems. The results were analyzed through the theoretical frame of Krutetskii (1976); Presmeg's (1985) mathematical visualization instrument and types of imagery, Guzman's (2002) types of visualization; and Donaldson's (1963) classification of errors. In short, the best demonstrated understanding was observed in the case of mathematical visualization preference and above average mathematical accuracy. Observed finding of the other two cases provide evidence of limitations in understanding surface areas. Students preferring analytical solutions struggled with graphing, students who preferred visual mathematical solutions with below average mathematical accuracy showed deviations from traditional understanding of basic shapes. A "left" cylinder and a star-shaped prism are examples of such discrepancy. This study is the beginning of research of how students learn multivariate calculus and what specific struggles they encounter. More research is encouraged to follow up on trends that emerged in this study.
Identifier: FSU_migr_etd-2512 (IID)
Submitted Note: A Dissertation submitted to the School of Teacher Education in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Degree Awarded: Fall Semester, 2008.
Date of Defense: November 6, 2008.
Keywords: Analytic Understanding, Visual Understanding, Multivariate Calculus, Students' Learning
Bibliography Note: Includes bibliographical references.
Advisory Committee: Kenneth Shaw, Professor Directing Dissertation; Sam Huckaba, Outside Committee Member; Leslie Aspinwall, Committee Member; Elizabeth Jakubowski, Committee Member.
Subject(s): Education
Persistent Link to This Record: http://purl.flvc.org/fsu/fd/FSU_migr_etd-2512
Use and Reproduction: This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s). The copyright in theses and dissertations completed at Florida State University is held by the students who author them.
Host Institution: FSU

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Meadows, Y. A. (2008). Calculus III Students' Analytic and Visual Understanding of Surface Areas of Spheres, Cylinders, Pyramids and Prisms. Retrieved from http://purl.flvc.org/fsu/fd/FSU_migr_etd-2512