Analysis of Electrical Engineering Students' Errors

Study Case Analysis of Electrical Engineering Students' Errors in Applying Chain Rules to Calculus Derivatives Michael Kurniawan (2018123456) University of Cincinnati, 2600 Clifton Ave, Cincinnati, OH 45221, United States kurniaml@mail.uc.edu Abstract Calculus is a compulsory course in Electrical Engineering programs and is an important prerequisite for taking advanced courses in the following semester. However, students still often have difficulty understanding the concept of derivatives, even though they have studied this material before. This study aims to identify various types of errors made by students in Calculus courses, particularly on the topic of derivatives. This study uses a qualitative descriptive approach involving 26 participants using purposive sampling techniques, as well as interviews with 6 students. After being given three essay questions, an analysis of the data was conducted to identify the types of errors that arose. The results show that conceptual errors are the most dominant type of error, accounting for 76.92% in the first question, 65.38% in the second question, and 61.54% in the third question. These errors occurred because students did not master various derivative formulas and lacked the habit of practicing solving problems to improve their problem-solving skills. Other contributing factors were low initial ability and a lack of interest in studying calculus. Keywords: Error Analysis, Electrical Engineering Students, Derivative Material. INTRODUCTION Mathematics is a field of study that must be learned from elementary school to university. Through mathematics education, students are expected to develop systematic, critical, logical, effective, and creative thinking skills, as well as honesty, discipline, and reasoning skills in problem solving. However, in reality, mathematics is still considered an uninteresting subject for MICHAEL KURNIAWAN 1 Study Case most students due to low learning outcomes. This condition can be caused by internal and external factors. One of the main causes is difficulty in understanding mathematical objects such as constructing mathematical models, determining and using the right formulas. As a result, the ability to understand mathematical concepts, principles, and skills becomes an important indicator in assessing a person's success in learning mathematics [1]. Based on the researcher's experience in teaching the topic of Derivatives in a Calculus course at the University of Cincinnati, many students make mistakes in solving mathematical problems. When students do not master basic mathematical concepts, they tend to make mistakes when working on Calculus problems, especially in understanding and applying the concept of derivatives [2]. In fact, most students have already studied derivatives in high school. Observations and interviews show that some students feel they have never learned derivatives, have forgotten the material, or did not pay attention to the teacher's previous explanations. This causes them to make similar mistakes at the university level [1]. Calculus itself is a basic course that must be mastered by Electrical Engineering students, because its concepts are used in various fields such as computer graphics lighting, artificial intelligence-based software development, and network systems [3]. With these many benefits, it is important for Electrical Engineering students to understand Calculus well. Unfortunately, low interest in learning and poor basic mathematical skills result in suboptimal learning outcomes in Calculus. Some students believe that Calculus is irrelevant to Electrical Engineering, and they have weak initial skills in algebra [4]. Based on Marzano's cognitive model, the biggest learning difficulty faced by students lies in the process monitoring level, namely difficulty in understanding abstract concepts, remembering prerequisite material, manipulating mathematical symbols, and performing integration and differentiation [5]. Therefore, it is important for lecturers to analyze the mathematical errors made by students so that they can be used as a basis for evaluating their mastery of Calculus material [6]. Based on the above description, this study aims to identify and describe various errors made by Electrical Engineering students in solving Calculus problems, particularly on the topic of derivatives using the chain rule. This analysis also aims to find the causes of errors and formulate more effective learning strategies for the next academic year. MICHAEL KURNIAWAN 2 Study Case The types of errors identified refer to Kastolan's classification, including conceptual errors, procedural errors, and computational errors. The research question in this study is: how can we describe the mathematical errors made by Electrical Engineering students at the University of Cincinnati in solving Calculus problems on the topic of derivatives using the chain rule? RESEARCH METHOD This study uses a qualitative descriptive approach, involving 26 first-semester Electrical Engineering students in the morning class at the University of Cincinnati who have completed the Basic Calculus course. The research subjects were selected using purposive sampling. The research object included various types of errors made by students in solving derivative problems in Basic Calculus material. The research began by compiling three derivative problems based on modifications from the Calculus book (Stewart, 2016). The time allowed for completion was 60 minutes without the use of books, the internet, cell phones, or laptops. Students were asked to complete three problems focusing on the chain rule: 1. Determining the derivative of inverse trigonometric functions, 2. Determining the derivative of natural logarithm functions, 3. Determining the natural logarithm derivative of trigonometric functions. The test results were analyzed and categorized based on the type of error, which was then presented in tabular form. From each error category, two students were selected for further interviews to explore the causes of the errors made. The research instruments consisted of a written test and semi-structured interviews with six students with different types of errors. Quantitative data analysis was performed using descriptive statistics in the form of percentages to determine the proportion of each type of error. The formula used was as follows: 𝑃𝑖 = 𝑛𝑖 𝑁 Explanation: Pi: Percentage of each type of error ni: Number of errors in each category N: Total number of errors identified MICHAEL KURNIAWAN 3 Study Case RESULTS AND DISCUSSION The initial stage of the research began with giving three essay questions about derivatives to 26 students as research subjects. Their answers were then analyzed to identify various types of errors that occurred. A total of 57.69% (15 students) completed all three questions, 26.92% (7 students) answered only two questions, 11.54% (3 students) answered only one question, and 3.85% (1 student) did not answer any questions. Figure 1. Bar Chart of the Number of Students Who Answered the Questions Next, the types of errors made by students in completing the questions were identified. The results showed that 34.62% (9 students) made conceptual errors in all three questions, 23.08% (6 students) made conceptual errors in two of the questions they answered, and there was 1 student (3.85%) who made conceptual errors in two questions but managed to answer one question correctly. A total of 15.38% (4 students) made conceptual errors on two questions and procedural errors on one other question. Three students (11.54%) only made one conceptual error on one question. Meanwhile, 1 student (3.85%) made a combination of conceptual and procedural errors, and the same number was also found among students who made conceptual, procedural, and computational errors at the same time. From the identification results, conceptual errors were the most dominant type of error among all research participants. Error Analysis for Each Question Question 1 — Derivatives of Trigonometric Inverse Functions A total of 22 students answered this question, and 76.92% (20 students) made conceptual errors, while 7.69% (2 students) made procedural errors. No computational errors were found in this question. MICHAEL KURNIAWAN 4 Study Case The most common conceptual error was the students' inability to write the derivative of the function 〖sec⁡〗^(-1) u. Figure 2. Answer to Question No. 1 by Student MVBM In Figure 2, the student with the initials MVBM made a conceptual error by incorrectly writing the derivative of 〖sec⁡〗^(-1) u. Based on an interview, he admitted that he did not memorize the derivative formula and only guessed its form, resulting in a conceptual error. A similar error also occurred in student MZ, as seen in the following example: Figure 3. Answer to Question No. 1 by Student MZ Student MZ correctly wrote the derivative d/dx(〖sec⁡〗^(-1) u)=1/(∣u∣√(u^2-1)) du/dx, but added the derivative formula for the multiplication of two functions, which was irrelevant in the context of the question. The interview revealed that MZ did not have a good grasp of the derivative formula for inverse trigonometric functions, resulting in a conceptual error even though part of the answer was correct. Second Question — Derivatives of Trigonometric Functions in Quotient Form This question required students to use the quotient rule and then the chain rule. A total of 65.38% (17 students) made conceptual errors, 7.69% (2 students) made procedural errors, and 3.85% (1 student) made computational errors. MICHAEL KURNIAWAN 5 Study Case Figure 4. Student AKS's Answer to Question No. 2 Student AKS made a computational error because he forgot to write the negative sign in the final result. Although his calculation steps were correct, this simple mistake affected the final result. In an interview, AKS realized his mistake and was able to explain the process again correctly. In addition, student NS also made a mistake on the second question, as shown in the following example. Figure 5. Answer to Question No. 2 by Student NS Student NS made a mistake in deriving the cosine function and incorrectly applied the quotient rule. Although the solution process he wrote was quite long, the steps were not relevant to the question. In an interview, NS admitted to writing down whatever he remembered without being sure of its correctness. Third Question — Derivatives of Natural Logarithms of Trigonometric Functions A total of 61.54% (16 students) made conceptual errors, 15.38% (4 students) made procedural errors, and no computational errors were found. MICHAEL KURNIAWAN 6 Study Case Figure 6. Answer to Question No. 3 by Student NIA Student NIA made a procedural error by not completely differentiating the exponential function. Although the steps to solve the problem were quite clear, this error caused the final result to be inaccurate. In an interview, NIA admitted to rushing through the question and missing that part. Figure 7. Answer to Question No. 3 by Student AAG Student AAG did not memorize the formulas for natural logarithm derivatives, trigonometric functions, or the chain rule. He also admitted that he did not understand the basic concepts even though he had tried to learn them. This indicates a conceptual error due to a weak grasp of basic calculus material. Figure 8. Bar Chart of Error Types for Each Question The diagram above shows that conceptual errors are the most dominant type of error in solving derivative problems. This reflects the low initial ability of students to understand basic mathematical concepts, especially calculus [7]. MICHAEL KURNIAWAN 7 Study Case Although most students have studied derivatives in high school, many of them did not pay attention to the teacher's explanations, so their understanding was not deeply formed. In addition, interviews revealed that most students did not utilize the online learning materials or videos provided by lecturers. They relied more on direct explanations in class and often copied their friends' work when doing assignments. Low learning independence and a lack of interest in mathematics further worsened learning outcomes [8]. Therefore, an initial ability assessment is necessary so that lecturers can adjust the level of difficulty of the material provided. In addition, students also need to be motivated about the importance of calculus in the field of electrical engineering so that they are more motivated to understand the concept of derivatives in depth. CONCLUSION Based on the results of the research and analysis that has been conducted, several conclusions can be drawn as follows: 1. Conceptual Errors Conceptual errors were the most common, occurring in the first question for 20 students (76.92%), the second question for 17 students (65.38%), and the third question for 16 students (61.54%). These errors were caused by a lack of mastery of various derivative formulas and a lack of practice with the questions. 2. Procedural Errors Procedural errors occurred in the first and second questions, each with 7.69% (2 students), and in the third question with 15.38% (4 students). These errors occurred because students did not write down the complete steps to solve the problem or forgot to apply one of the derivative formulas. 3. Computational Errors Computational errors were only found in one student on the second question (3.85%). This was due to a lack of precision in writing down the signs or double-checking the final calculation results. MICHAEL KURNIAWAN 8 Study Case REFERENCE MICHAEL KURNIAWAN 9