The concept of derivative is one of the fundamental and important concepts in calculus. It is a concept built from other concepts such as functions and limits (Zandieh 1998).
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Research on understanding derivative has shown that students have deficiencies in their conceptions of derivatives. More specifically, they face difficulties in
understanding the derivative as a rate of change (Bezuidenhout, 1998; Orton, 1983), and understanding the graphical representation of derivative as the slope of the tangent line (Orton, 1983, Ferrini- Mundy and Graham 1994; Asiala et al., 1997; Vinner 1982). Moreover, students face difficulties in relating the derivative function with the original function (Orhun, 2012; Ubuz, 2007) and understanding the formal definition of
derivative (Zandieh, 1998). Research has shown (Orton, 1983; Tall, 2011) that students' difficulties in learning derivative are due to their lack of the conceptual understanding of the concept.
Orton (1983) interviewed 110 college and precollege students to investigate their understanding of derivatives. Students were asked to perform some routine problems (finding derivative of functions using some derivatives rules) and do some conceptual tasks that include interpreting graphs, finding slopes graphically, grap hical interpretation of both average and instantaneous rate of change, etc. Orton found that almost all
students did well on the routine differentiation items such as finding the derivative of polynomial functions. However, he noticed that many students were not able to relate the derivative of a function at a point with the slope of the tangent line, nor to the limit of a set of secant lines. Other areas of difficulties are related to the ideas of instantaneous rate of change versus average rate of change. Orton also found that about 20% of
students got confused with the derivative at a point and the y-value of the point of tangency.
Ferrini- Mundy and Graham (1994) conducted a study to investigate college students understanding of different calculus topics including derivatives. All of the
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students were interviewed and asked to think aloud while completing a set of tasks some of which were presented graphically and others algebraically. Graham and Mundy discussed in details one student’s attempts to find the equation of a function presented graphically before sketching the graph of the derivative. Although this student was able to find the derivative of a function given its equation and sketch its curve, yet she had difficulties in other areas. She was unable to relate the derivative of a function to the tangent line, and she had no geometric meaning of differentiability. The researchers concluded, “Graphical contexts and algebraic contexts may function for students as separate worlds” (p.42).
Asiala et al. (1997) conducted a study to investigate the graphical understanding of a function and its derivative by university students. Forty -one students participated in the study where 17 students took a reformed calculus course and were taught using cooperative learning and computers, while the remaining took a traditional calculus course. Clinical interview conducted with each student consisted of eleven questions. Students were asked to justify their answers and reasoning while solving the questions. In one question, students were given only the graph of a function and a tangent line on some particular point, and they were asked to find the derivative at that point. Another question asked students to sketch the graph of a function based on some information given in a table form. Students’ responses were analyzed according to APOS levels. The results showed that students who received the instructional treatment given in the
reformed calculus course showed better understanding of the function and derivative concepts (at the process and object levels) than students in the control group.
In his study, Huang (2011) examined engineering students' conceptual understanding of the derivative concept. The sample of the study consisted of 35
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students. They were exposed to the derivative concept using different modes of
representation: symbolic (rules of derivatives), graphical and numerical (tables and rate of change). After the implementation of the unit on derivatives, a test composed of two test problems was given to students who were required to explain their thinking and problem- solving processes. Moreover, semi- structured interviews were conducted with some students. The results showed that almost 80 % of students' conception approached the object level. Another study that examined students' conceptual understanding of derivative, based on APOS, was that conducted by Maharaj (2013). A multiple-choice test composed of six questions was administered to 857 university students. Students were tested on applying the rules of derivatives and their applications (e.g. rate of change, interpretation the graph of a derivative and an optimization problem).The findings of the study revealed that students had troubles when applying the chain rule, solving the rate of change problem and interpreting the derivative of a function given graphically. The researcher concluded that the majority of students do not have adequate mental structures at the process, object and schema levels. It was suggested that more time should be devoted to instruction focusing on the numerical (rate of change) and graphical approaches of the derivative concept.
Concerning the formal definition of derivative, which includes the knowledge of the limit of the quotient difference or ratio, Zandieh (1998) conducted a study with nine high school students to explore their abilities to apply the definition in different
representations. All students were interviewed and asked different questions on the concept of derivative. As to students’ responses, they were classified into three categories. Some students showed a good understanding of the formal definition of derivative, three students had memorized the definition with no understanding and they
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were not able to relate it to the notions of limit, ratio and functions, while two of them did not memorize the definition nor did they understand it.