Analysis of Students' Cognitive Difficulties in Mastering

Study Case Analysis of Students' Cognitive Difficulties in Mastering the Concept of Function Derivatives and Its Implications for Calculus I Learning Michael Kurniawan (2018123456) University of Cincinnati, 2600 Clifton Ave, Cincinnati, OH 45221, United States kurniaml@mail.uc.edu Abstract The concept of function derivatives is one of the important topics in the Calculus course taught to students in the Electrical Engineering Program at the University of Cincinnati. This material plays a fundamental role, especially for engineering students, as it forms the basis for understanding various calculation concepts in advanced courses. However, many students still face difficulties in understanding the concept and solving problems related to this topic. This study aims to identify the level of difficulty students experience in mastering the concept of function derivatives and to find the factors that cause these difficulties. The research method used is descriptive qualitative, in which the researcher describes and explains in detail the problems experienced by students in learning function derivatives in Calculus courses. The research data was obtained from the quiz results of students in the Electrical Engineering Study Program for the 2024/2025 academic year. The results show that 69% of students made principle errors (KP), while 31% experienced calculation errors (KH). Based on these findings, it can be concluded that most students have difficulty remembering the relevant rules, formulas, and basic theorems, and are not yet able to identify concepts properly. The results of this study are expected to serve as a basis for the development of more interactive and interesting Calculus learning methods, thereby increasing students' enthusiasm and understanding of the material taught. Keywords: difficulty analysis, calculus, function derivatives. MICHAEL KURNIAWAN 1 Study Case INTRODUCTION The Calculus course plays a very important role in higher education, especially for students in science and technology. Calculus provides the mathematical foundations needed to analyze changes, optimization, and various real phenomena in everyday life. In engineering, Calculus has high practical value because almost all engineering disciplines use this concept to solve problems and design complex systems. According to [1], in the field of computer science, the concept of calculus is used in the development of artificial intelligence (AI) such as expert systems, computer games, fuzzy logic, neural networks, and robotics. Function derivatives are one of the main components in calculus that play an important role in technical analysis, for example, to measure the rate of change in electric current, voltage variations in circuits, and the optimization of structural or system designs. Therefore, electrical engineering students are required to thoroughly understand the basic theory of function derivatives. However, in reality, many students still have difficulty understanding and applying these concepts. Research conducted by [2] shows that by solving derivative problems, students can develop creative and systematic thinking skills, especially in determining problemsolving strategies. Nevertheless, many students still face difficulties in proving basic concepts and theorems, including an inability to use mathematical notation correctly and uncertainty in implementing effective proof approaches [3][4]. According to [5], the main difficulty in learning calculus lies in understanding basic concepts such as limits, derivatives, and integrals, which require a deep conceptual understanding and strong mathematical calculation skills. With a quantitative research approach to students taking calculus courses, this study emphasizes the importance of studying the analysis of difficulties in learning calculus to improve teaching strategies, conceptual understanding, and the effectiveness of learning materials and methods. Based on this background, this study focuses on analyzing the difficulties students face in mastering the concept of function derivatives in Calculus courses. The main objective is to improve the quality and effectiveness of learning, as well as to provide a basis for designing teaching methods that are appropriate for the characteristics of students at the University of Cincinnati. This study is expected to make a significant contribution to educational institutions and the development of mathematics learning in the United States. Similar studies on MICHAEL KURNIAWAN 2 Study Case students' difficulties in solving Calculus problems have also been conducted by several researchers [6]. RESEARCH METHOD This study uses a qualitative descriptive method. The main objective is to identify the level of difficulty students have in understanding the concept of trigonometric function limits and to find the factors causing these difficulties. The research subjects were students of the Electrical Engineering Study Program for the 2024/2025 academic year at the University of Cincinnati, with a total of 7 students participating. Data Collection Techniques Data was collected through tests and interviews. • The test was given in the form of a quiz containing several questions about function derivatives. • Interviews were conducted to obtain additional information about the reasons and thought processes of students in answering the questions on the test. Data Analysis Techniques Data analysis refers to the Miles and Huberman model, which includes three main stages: 1. Data Reduction, which is examining and analyzing students' answers by noting the types of errors that arise during the problem-solving process. 2. Data Presentation, which involves displaying the test and interview results in the form of descriptions, tables, and graphic illustrations. 3. Conclusion Drawing, which involves comparing the test results with the interview data to gain a comprehensive understanding of the patterns of errors that occurred. Test results are classified based on two main error categories, namely Principle Errors (PE) and Calculation Errors (CE). The indicators for each category are shown in Table 1 [1]. MICHAEL KURNIAWAN 3 Study Case Table 1. Categories and Indicators of Student Errors RESULTS AND DISCUSSION Results This section explains the results of the data collected through student tests (quizzes). The test consisted of three questions as follows: 𝑑𝑦 1. Determine 𝑑𝑥 from 𝑦 = (𝑥1 + 𝑥12 )(3𝑥 3 + 27) 𝑑𝑦 2. Determine 𝑑𝑥 from 𝑦 = 2𝑥 sin(𝑥) 𝑑𝑦 3. Determine 𝑑𝑥 from 𝑦 = 𝑥 2 𝑦 + sin⁡(𝑥𝑦) − 5𝑥 The research subjects were students of the Electrical Engineering Study Program class of 2024 at the University of Cincinnati, with a total of 13 students. The test was conducted on time, and all participants took part in the activity in an orderly manner. After all answer sheets were collected, the researcher selected subjects who met the research criteria, namely students who showed a high and varied level of errors in answering questions. Some students made mistakes on the same points, while others made mistakes on different parts. The list of students selected as research subjects is shown in Table 2. Table 2. List of Research Subjects MICHAEL KURNIAWAN 4 Study Case Discussion Based on the analysis of the six students who were the research subjects, the following is a detailed explanation of the results of the review of the students' quiz answers on function derivatives. Subject ACH Based on Figure 1, the results of subject ACH's work on question number 1 show a fundamental error, in which the student was unable to identify the relevant elements in the function, resulting in an error in the calculation. The subject should have first identified the variables u and v and their respective derivatives systematically, then completed the calculation steps gradually. In question number 2, the subject also made a fundamental error in the element identification stage, even though the calculation process was correct. The identification steps should be explained in a more structured manner, as in the correction in question number 1. Meanwhile, the work on question number 3 was correct. Figure 1. Results of subject ACH's work. Subject BAS Based on the results in Figure 2, subject BAS showed correct results in question number 1. However, in question number 2, the student made a calculation error in deriving the sin⁡(x) function. For question number 3, which was related to the application of the chain rule, the results were correct. MICHAEL KURNIAWAN 5 Study Case Figure 2. Results of subject BAS's work. Subject SAG From Figure 3, it can be seen that subject SAG made a calculation error in question number 1 when deriving the multiplication function. In question number 2, SAG made a fundamental error in identifying the variables u and v and their derivatives, even though the final calculation result was correct. Meanwhile, question number 3 was solved correctly. Figure 3. Results of subject SAG's work Subject AP Based on Figure 4, subject AP made a calculation error in question number 1, particularly in the process of deriving the multiplication function and its explanation. In question number 2, subject AP made a fundamental error in the identification of the variables udan and v, even though the final calculation result was correct. For question number 3, the steps to solve the problem were explained well and the final result was correct. MICHAEL KURNIAWAN 6 Study Case Figure 4. Subject AP's work results Subject LM Subject LM's work results (see Figure 5) show errors in all questions. In question number 1, there was a fundamental error because the student did not recall the basic formula for the derivative of a multiplication function and failed to identify the concept correctly. In question number 2, there was a calculation error in applying the derivative of a multiplication function. Meanwhile, in question number 3, the student again made a fundamental error in deriving the division function, resulting in an incorrect solution procedure. Figure 5. Results of subject LM's work. Subject MDA Based on Figure 6, subject MDA made fundamental errors in all questions (numbers 1–3). This occurred because the student could not recall the formulas related to function derivatives. As a result, the steps taken to solve the problems were incorrect, MICHAEL KURNIAWAN 7 Study Case and the calculation results were wrong. It can be concluded that subject MDA did not fully understand the basic concepts of function derivatives. Figure 6. Results of subject MDA's work. Based on the analysis results of all research subjects, a recapitulation of the results and error categories is presented in Tables 3 and 4 below. Table 3. Assessment Results of Research Subjects on Each Question. MICHAEL KURNIAWAN 8 Study Case Figure 7. Graph of Student Error Rates for Each Question Based on the graph in Figure 7, it can be seen that the lowest error rate occurred in question number 3, which was related to the application of the chain rule. Conversely, the highest error rate appears in question number 2, which involves the derivative of a product function. Table 4. Categories of Research Subject Errors on Each Question Explanation: KP = Principle Error MICHAEL KURNIAWAN KH = Calculation Error 9 Study Case Figure 8. Graph of the Percentage of Error Categories of Research Subjects From the analysis in Figures 7 and 8, it can be concluded that the research subjects made the fewest errors on question number 3 and the most errors on question number 2. Overall, 69% of students made Principle Errors (KP), while 31% made Calculation Errors (KH). This shows that most students had difficulty remembering basic rules, formulas, or theorems, and were unable to correctly identify material concepts. These findings are in line with previous studies [6][7], which show that students' difficulties in understanding function derivatives are generally caused by a lack of mastery of basic concepts, errors in applying formulas, and weak abilities in starting calculation steps systematically. In addition, other studies [8][9][10][11] also show that students often have difficulty understanding the concepts of differential calculus, particularly in the topics of infinite series, limits, and derivatives, which affects their ability to determine the rate of change and extreme points of functions. Further research [12] indicates that student errors generally occur at the encoding, process skill, and transformation stages, which shows a weak understanding of basic procedures and concepts. To overcome these problems, research by [13] emphasizes the importance of designing learning strategies that take into account students' learning obstacles, whether conceptual, procedural, or psychological, so that calculus learning can be more effective and meaningful. MICHAEL KURNIAWAN 10 Study Case CONCLUSION Based on the results and discussion of this study, it can be concluded that students' difficulties in mastering the concept of function derivatives in Calculus I can be divided into two main categories, namely Principle Errors (PE) and Calculation Errors (CE). The analysis shows that 69% of students made Principle Errors (PE), while 31% made Calculation Errors (CE). This indicates that the main cause of students' difficulties lies in their inability to remember the rules, formulas, or theorems they have learned and their lack of ability to identify the concepts in the material accurately. These findings are expected to serve as a basis for lecturers and educators in designing more interactive and contextual learning models, for example through the use of digital media, mathematical simulations, or project-based learning. Thus, it is hoped that students will become more interested, active, and motivated in studying Calculus I, especially on the topic of function derivatives. REFERENCE MICHAEL KURNIAWAN 11 Study Case MICHAEL KURNIAWAN 12