Available online at www.sciencedirect.com
Procedia Social and Behavioral Sciences 1 (2009) 1792–1796
World Conference on Educational Sciences 2009
Misconception in Calculus-I: Engineering students’ misconceptions in the process of finding domain of functions Erdo÷an M.Özkana, Hasan Ünala a
Yildiz Technical University,østanbul,Turkey
Received October 25, 2008; revised December 23, 2008; accepted January 5, 2009
Abstract The purpose of this study was to investigate students’ misconceptions in the process of finding domain of a composite function. The participants were 1500 engineering students in a state university in Istanbul, Turkey. The Calculus course is required course for all engineering students. Data analysis revealed that, students, misconceptions fall into five categories. First category was domain and ranges are mixed up. Second Category was students used derivative of function. Third category was using union of functions instead of intersection of domain of each function. Fourth category was application of delta-epsilon technique. Fifth and final category was irrational function and denominator. © 2009 Elsevier Ltd. Open access under CC BY-NC-ND license. Keywords: misconcetions in mathematics; domain; functions,calculus
1. Introduction First year in college is a significant time in students’ mathematical understanding and development in which they start to crystallize their understanding of mathematical concepts and to see engineering application of problems in various university mathematics courses starting from Calculus I. At this stage students also get confuse due to unnoticed misconception built upon their high school mathematics. The objective of this study to create an informative resources for University Instructors about common misconception exhibited by first year engineering students’ finding domain of a composite function. Misconception as research domain has gained all areas of school subjects (ie.Biology, Barrass, 1984 , Statistics, Hand, 1998). The limit concept is fundamental to understanding of calculus. According to Tall(1992) ‘ The idea of limit signifies a progression to a higher level of mathematical thinking’ (p. 495).Other concepts like derivative or integration( i.e. Riemann sum) is basically application of limit concept. The number of studies ( Davis and Vinner, 1986 ; Tall, 1992 ; Jordan, 2005) has shown that students’ understanding of limit concept is problematic area, Jordan(2005) list the potential reasons that cause the problem. One of the such reasons is that ‘ ..the idea of a limit as the first mathematical concept that students meet where one does not find a definite answer ‘(p.21). Second
[email protected] and
[email protected]
1877-0428 © 2009 Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.sbspro.2009.01.317
Erdog˘an M.Özkanet al. / Procedia Social and Behavioral Sciences 1 (2009) 1792–1796
1793
potential reason was related to the concept of limit being abstract and students abstraction levels. Another reason might be conceptual v.s. procedural understanding. In the recent study Unal and Ozkan (2008) explored the engineering students misconception about the application of L’ hospital rule. They found out students forms four main type misconception when they find the limit using L’ hospital rule. 2. Study Participants were engineering students (1500) in a state university in Istanbul, Turkey. For the purpose of this study students were asked to find the domain of function problem with written explanations. Data matrix built based on the following categories: True answer, blank paper, and solutions that carry misconceptions. Data were analyzed jointly by the researchers. Followings were the problem used in this study Find the domain of f ( x) = sin(ln x) +
tan x 1− x
Composite function f ( x) which was consists of: Trigonometric functions, natural logarithmic function, irrational function, and absolute value function. From the data analysis five main type of misconception has been identified. First category was domain and ranges are mixed up. Second Category was students used derivative of function. Third category was using union of functions instead of intersection of domain of each function. Fourth category was application of delta-epsilon technique. Fifth and final category was irrational function and denominator. In the following section samples were given:
1794
Erdog˘an M.Özkanet al. / Procedia Social and Behavioral Sciences 1 (2009) 1792–1796
Figure 1: Second type misconception
As it is shown in the Figure 1 students were trying to find the domain of function by using the derivative of the function.
Figure 2: First type misconception
In the first category students mixed by the definition of domain and range as shown in the Figure 2 above.
Erdog˘an M.Özkanet al. / Procedia Social and Behavioral Sciences 1 (2009) 1792–1796
1795
Figure 3: Union of domain
Students used union of each domain. The intersection of domain should have been used.
Figure 4: Fourth type of misconception
In this type of misconception students used delta-epsilon technique which was subject of definition of limit.
Figure 5: Final misconception
In this type of misconception students just concern about the rational functions. However for the domain characteristics of each function should have been investigated. The purpose of this study was to identify the misconception in the process of finding domain of functions. The analysis revealed that five main types of misconceptions occur. For each type of misconception as an educators we need to find cure for each type misconception. Reference Barrass, R. (1984) Some Misconceptions and Misunderstandings Perpetuated by Teachers and Textbooks of Biology. Journal Of Biology Education 18 : 201-205. Davis R. & Vinner S. (1986) The Notion of Limit. Some Seemingly Unavoidable Misconception Stages. Journal of Mathematical Behaviour , 5, pp. 281-303. Hand D. J. (1998) Breaking misconceptions - statistics and its relationship to mathematics. Journal of the Royal Statistical Society, Series D, The Statistician, 47, 245—250
1796
Erdog˘an M.Özkanet al. / Procedia Social and Behavioral Sciences 1 (2009) 1792–1796
Jordan, T.(2005). Misconceptions of the limit concept in a Mathematics course for Engineering students. Unpublished Master Thesis, University of South Africa Tall, D. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity, and proof. In D. A. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 495-511). New York: MacMillan Publishing Unal H, and Ozkan, E.M. (2008). Misconception in calculus course for engineering students: The case of finding limits, The paper presented at the 9th annual meeting further education in the balkan countries, Konya, Turkey
