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# Engineering students' conceptual understanding of the derivative in calculus

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## Abstract

The derivative is one of the important topics that engineering students encounter at university level. The research reported here, applied a theoretical framework combining the strengths of two major mathematics education theories in order to investigate the learning and teaching of the derivative. This article highlights some of the overall findings of this research with regard to students' understanding of the concept of derivative, and suggests applications of the framework to learning and teaching in undergraduate mathematics classrooms.

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... In addition, in some studies, focus has been on students' mental constructions of the derivative concept in different representations (Borji et al., 2018;Huang, 2011;Tokgoz, 2012). The derivative concept can take on geometric (graphical), algebraic, numerical and verbal representations, since the concept is directly related to functions. ...
... Firstly, some students were not able to correctly link the derivative at a point and the slope of the tangent line at the same point. Huang (2011) noted that the difficulties experienced in this category may be because students had not yet constructed the function concept as a process. The students in the second category displayed some degree of understanding but were unable to describe the equality of the two; hence, they used the tangent equation to find the slope of the tangent line. ...
... The findings of their study showed that all students from the control group (except for two) and seven students from the experimental group did not know the relationship between the limit of the difference quotient and the value of the derivative at a point. A similar problem was encountered with the students in the study by Huang (2011). This high success of the experimental group is attributed to the use of computer technology. ...
Article
The derivative is a central concept in calculus and has applications in many disciplines. This study explored students’ understanding of derivatives with a particular focus on the graphical (geometric) representation. The participants were four Mathematics Honours students from a university in Lesotho. Data were generated from the written responses to five tasks and interviews, with all collection methods included. The Action–Process–Object–Schema (APOS) theory was used as the framework of analysis. The findings of the study show that the mental constructions that students developed in learning the derivative concept were mainly actions. The coordinated process conception of the derivative as the limit was evident with respect to algebraic representation but not in the graphical representation. Only one student showed that they had the process conception of the secant lines tending to the tangent line. It was also found that the process of tending to the limit operator was interpreted as tending to and landing on rather than being in the neighbourhood of by some students. These findings suggest that students should be made aware of the interpretation of tending to as implying staying in the neighbourhood rather than landing on. Students should also be given activities based on derivatives that require translation from one mode of representation to another (algebraic to graphical in this case).
... NCTM (2000) suggests that the term representation refers to processes and products that should be viewed as essential elements to support an individual's understanding of mathematical concepts. The derivative concepts can be represented by graphically as the slope of the tangent line to a curve at a point or as the slope of the line a curve seems to approach under magnification, verbally as the instantaneous rate of change, physically as speed or velocity, and symbolically as the limit of the difference quotient (Borji et al., 2018;Huang, 2011;Tokgöz, 2012). Outline of the framework for exploring multiple representations of derivative concepts as presented by Zandieh (2000) is presented in Figure 1. ...
... The review is an attempt to organize teaching about derivative concepts to be more comprehensive and meaningful. Zandieh's statement is supported by several studies that focus on mental construction, where the results of these studies confirm that participants show different representations in learning derivative concepts (Borji et al., 2018;Huang, 2011;Tokgöz, 2012;Moru, 2020). According to Duval (2006) and Bressoud (2016), research related to the construction of cognitive structures refers to a review the notion of concept image and concept definition as a theoretical framework for analyzing research findings. ...
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Derivative concept is one of the essential studies in calculus, which is studied in teaching mathematics. Prospective mathematics teachers who have completed their studies and later become teachers will teach derivative concepts to their students at school. Therefore, knowledge of derivative concepts is vital in transforming knowledge to students. This study aimed to investigate concept images of prospective mathematics teachers on derivative representations. The research design in this study used a qualitative with case study approach. The participants were prospective mathematics teachers at a university in West Java, Indonesia (N=29). The research data was obtained from the test and clinical interview. The findings of this study show that the concept image of all participants on the derivative concept is still limited in function representation. Concerning the meaning of the derivative concept, most participants only view the derivative concept as a tool to solve procedural problems. It concluded that the representation of participants still did not support conceptual understanding of the derivative concept. It is the impact of the teaching design that given. Based on these findings, educators are expected to be able to improve the quality of teaching derivative concepts in the future by using various contexts or representations so that the concept image formed is more comprehensive to support conceptual understanding in learning of derivative concepts.
... Then, they introduced derivatives as the slope of the tangent line in a specific point on the function, and they concluded by presenting the rules of computing derivatives. In [36], it was shown that students would succeed in understanding the concept of derivatives if they were able to successfully develop its different definitions and representations, i.e., the rate of change, the slope of the tangent line, the limit, the rules. ...
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This study presents a designated flipped classroom (FC) mathematics environment that utilizes a unique online platform designed for Arab minority students in Israel. It investigates how studying in an FC affects conceptual understanding and motivation to study mathematics among Arab high school students. The study also explores the factors that contribute to effective learning in the FC environment. Participants were 75 Arab high school students in 10th and 11th grades who studied advanced mathematics. Each grade group was randomly divided into two subgroups: an FC group and a traditional classroom group (comparison group). Quantitative questionnaires given before and after the learning program served to measure students’ motivation and conceptual understanding of the derivative and integral topics. Additionally, a random sample of students who studied in the FC group and the teacher who taught all the groups were interviewed. The study describes the positive effect an FC environment has on students’ conceptual understanding, particularly for 11th graders. The participants mostly appreciated how the FC resulted in less lecturing in class. The study contributes to the literature about FC among minorities and contributes to national and international efforts being made to reduce the gap in mathematics achievements between minorities and other sectors.
... As efforts proceed toward a deeper understanding the cognitive processes associated with comprehension of differential topics in 3D, it is important to recognize the importance of the semiotic elements discussed in this article that are currently often overlooked or ignored. As was the case in the work of Huang (2011) with 2D derivatives, an initial effort to use these semiotic elements is being explored by one of the authors using Action-Process-Objects-Schema (APOS) Theory (Arnon et al., 2013). APOS Theory requires a "genetic decomposition" of cognitive processes in order to describe specific mental constructions that students make when understanding mathematical concepts. ...
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In two dimensions (2D), representations associated with slopes are seen in numerous forms before representations associated with derivatives are presented. These include the slope between two points and the constant slope of a linear function of a single variable. In almost all multivariable calculus textbooks, however, the first discussion of slopes in three dimensions (3D) is seen with the introduction of partial derivatives. The nature of the discussions indicates that authors seem to assume that students are able to naturally extend the concept of a 2D slope to 3D and correspondingly it is not necessary to explicitly present slopes in 3D. This article presents results comparing students that do not explicitly discuss slopes in 3D with students that explicitly discuss slopes in 3D as a precursor to discussing derivatives in 3D. The results indicate that students may, in fact, have significant difficulty extending the concept of a 2D slope to a 3D slope. And that the explicit presentation of slopes in 3D as a precursor to the presentation of derivatives in 3D may significantly improve student comprehension of topics of differentiation in multivariable calculus.
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ARTICLE INFO ABSTRACT Derivative is one of the most important topics in calculus that has many applications in various sciences. However, according to the research, students do not have a deep understanding of the concept of derivative and they often have misconceptions. The present study aimed to investigate undergraduate basic sciences and engineering students' understanding of the concept of derivative at Tehran universities on based the framework of Zandieh. The method was descriptive-survey. The population included all undergraduate students of Tehran universities who passed Calculus I. The sample included 604 students being selected through multi-stage random cluster sampling. The measurement tool was a researcher-made test for which the reliability coefficient was obtained using Cronbach's alpha (r=.88). Inspired by Hähkiöniemi's research, nine tasks on derivative learning were given to the students. The students' responses were evaluated using a five-point Likert scale and analyzed using descriptive responses. The results indicated that students have no appropriate understanding of the basic concepts of derivatives in numerical, physical, verbal, and graphical contexts. Basic sciences students performed meaningfully were better in understanding the tangent line slope compared to engineering students, while engineering students performed meaningfully were better than basic sciences students in the rate of change.
Article
Investigates the extent to which visual considerations in calculus can be taught and be a natural part of college students' mathematical thinking. Recommends that the legitimacy of the visual approach in proofs and problem solving should be emphasized and that the visual interpretations of algebraic notions should be taught. (YP)