# The Erroneous Derivative Examples of Eleventh Grade Students

## Article excerpt

Abstract

The derivative is not only an important subject for mathematics but also is an important subject for engineering, physics, economy, chemistry, and statistics. Especially, mathematics depends on strongly preceding learning and the subject of derivative will be used in university education by all students. Therefore, it is one of the most important subjects. This study's purpose is to explore student mistakes and errors in derivative and determine the areas in which students have probable misconceptions. For this purpose, 7 questions were chosen from "the Student Placement Test" (OSS). These questions were transferred into open-ended questions. The results of the study took place at sixth form college are described and discussed. The test administered to 53 students from Balikesir Fatma Emin Kutvar Anatolian High School in the fall-term of 2005-2006. Determining the possible misconceptions should help teachers when they teach this subject. The study findings showed that students could not understand derivative definition that depends on limit, make mistakes in composite functions and trigonometric functions, and establish wrong relations between tangent's slope, and normal's slope. Teachers need to be able to find errors and misconceptions in students' solutions. Teachers also need to be applying meaningful learning strategies such as concept maps, worksheets about derivative (e.g. Appendix B, Appendix C).

Key Words

Errors and Misunderstanding, Misconceptions, Errors and Misunderstanding Towards Derivative

(ProQuest: ... denotes formulae omitted.)

The derivative of a function represents an infinitesimal change in the function with respect to whatever parameters it may have. The "simple" derivative of a function f with respect to x is denoted either f(x). Students have some misconceptions or errors in derivative. Misconception is defined as erroneous conception, false opinion, or wrong understanding (Big Larousse, 1986). Studies about derivative and ideas related to it (such as tangent lines) have emphasized students' misconceptions and common errors (e.g., Amit & Vinner, 1990; Artique, 1991; Orton, 1983; Ubuz, 1996, 2001; Maurer (1987; Norman & Pritchard, 1994; Krutetski, 1980; Orton, 1983; Donaldson, 1963; Cipra, 1989; Keith et al., 1990). Ubuz (2001, p. 129) showed that students' common misconceptions on derivative were as follows: " (a) derivative at a point gives the function of a derivative, (b) tangent equation is the derivative function, (c) derivative at a point is the tangent equation, and (d) derivative at a point is the value of the tangent equation at that point." Ubuz also found that students seem to think different concepts as the same. He reported that "(a) the lack of discrimination of concepts which occur in the same context or the confusion of a concept with another concept describing a different feature of the same situation, (b) the inappropriate extension of a specific case to a general case, and (c) the lack of understanding of graphical representation"(p.133). Some studies have mainly focused on the constructions of mathematical knowledge in a theoretical perspective rather than students' misconceptions and common errors (Dubinsky & Schwingendorf, 1991) On the other hand, few empirical research were conducted such as Tall (1986a). He revealed that 67% of the experimental students who used Graphic Calculus (Tall, 1986b) chose the right answer with a correct explanation, while only 8% of the control students did. Thus, it is likely that visualization in the graphical context can help students understand the relations between differentiation and integration. Mathematics teaching is directly linked to learning and students' understanding of the concept of derivative is related to their prior knowledge (Kendal, 2001). Kendal (2001) stated that using multiple presentations was important in developing the understanding of the concept of derivative. In the present study the following questions are addressed. …