Type of Document Dissertation Author Meadows, Yelena A. Author's Email Address email@example.com URN etd-11102008-162937 Title Calculus III Students' Analytic And Visual Understanding Of Spheres, Cylinders, Pyramids And Prisms. Degree Doctor of Philosophy Department Teacher Education, School of Advisory Committee
Advisor Name Title Ken Shaw Committee Chair Elizabeth Jakubowski Committee Member Leslie Aspinwall Committee Member Sam Huckaba Outside Committee Member Keywords
- Analytic Understanding
- Visual Understanding
- Multivariate Calculus
- Students' Learning
Date of Defense 2008-11-06 Availability unrestricted AbstractCalculus 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.
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